a. Estimating vertical air motion from power spectra

The process of retrieving the vertical air velocity used in this paper was first developed by Probert-Jones and Harper (1961) and was adapted by Battan and Theiss (1966). The LBM (Atlas 1964, p. 413; Atlas et al. 1973, p. 28) assumes that the smallest particles detectable by radar have the smallest terminal velocity and are represented as the most positive velocity in the power spectra that is above the noise level (the “smallest particles” point in Fig. 1). Atlas et al. (1973, p. 29) note that “the lower-bound method should work quite well in the case of snow.” The method of deriving the vertical air motion from Doppler spectra in this study was adapted from Battan (1973) to account for the effects of spectral broadening resulting from the mean horizontal wind, wind shear, and turbulence (Donaldson and Wexler 1969; Atlas 1964, p. 413).

For a 915-MHz radar, contributions to the total backscatter power (and hence the Doppler spectrum) may arise both from Rayleigh scatter from raindrops and ice particles and from Bragg scatter associated with refractive index inhomogeneities. The LBM can retrieve vertical motion under the condition that the Doppler spectra are produced only from Rayleigh scatter. Bragg scatter contributions, if present, would produce a positive bias in vertical motion values retrieved with the LBM. In the appendix, we show that contributions from Bragg scatter are negligible for the cases examined.

The LBM attempts to determine the radial velocity of the smallest detectable particles within the radar beam. The smallest particles have the lowest terminal velocity and therefore can be assumed to follow the air motion most closely. The smallest detectable particles are represented in the Doppler power spectra at the rightmost point above the noise level (Fig. 1). The first step in the LBM is to determine the radial velocity V at this point.

The mean noise level was determined by calculating the mean of the power in the velocity intervals from +8 to +18.16 m s⁻¹ (the positive Nyquist velocity) and −6 to −18.16 m s⁻¹ (Fig. 1). These velocity intervals were completely outside the range with powers above the noise level for this dataset. The maximum V with a power value at least two standard deviations above the mean noise level was then determined using the following algorithm. The peak of the Doppler spectrum was first identified (Fig. 1). Next, the data point to the right of the peak of the Doppler spectrum that fell below a power value of two standard deviations above the mean noise power level was identified. The velocity at the point to the immediate left of this data point was assigned as V and was used in subsequent calculations to estimate the vertical air velocity using the LBM, with one exception. The MIPS algorithm for acquiring data included ground-clutter suppression, which effectively reduced the power at 0 m s⁻¹ to a value below the mean noise in the power spectrum. If the first point to the right of the peak of the Doppler spectrum was at 0 m s⁻¹, the next point to the right was also examined. If the power at this point was greater than two standard deviations above the mean noise power level, then the noise point at 0 m s⁻¹ was ignored in the determination of V.

Although the particles contributing to the power at velocity V are the smallest and most closely follow the air motion, they have a nonzero terminal velocity. To estimate w, their terminal velocity must be taken into account. There were no aircraft measurements of particle size distributions or habits collected within the precipitation bands during this study. To estimate particle terminal velocities, measurements taken from similar bands in a winter storm reported by Passarelli (1978) were used.

Figure 2 shows particle size spectra at four flight altitudes sampled by Passarelli during a period of moderate to heavy snowfall over Champaign, Illinois, on 26 November 1975. The spectra were derived from data collected with an optical array precipitation spectrometer. Passarelli, using a precipitation particle camera, primarily observed aggregates of side plane–type crystals at the 4.35-km (−20°C) and 3.75-km (−17°C) altitudes and dendritic-type aggregates at the 3.15-km (−15°C) and 2.55-km (−12°C) altitudes. Terminal velocity–diameter relationships developed by Locatelli and Hobbs (1974) for side plane–type aggregates,

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and dendritic-type aggregates,

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with the density correction factor from Foote and du Toit (1969),

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were applied to estimate the fall velocities of the ice particles observed by Passarelli. Data reported by Locatelli and Hobbs (1974) were used to calculate the air density ρ₀ at their measurement site, and the nearest sounding to the profiler was used to calculate the air density ρ at the four altitudes corresponding to the Passarelli study.

The reflectivity-weighted contribution of particles to the terminal fall velocity within the size bin ΔD,

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was then determined for each of the size spectra in Passarelli (Fig. 3). Figure 3 shows that the smallest-sized ice particles that had a nonnegligible contribution to the power in the Doppler spectrum ranged in diameter from 0.5 to 0.9 cm. The curves in the upper part of the figure show the terminal velocities of the particles at the four altitudes observed by Passarelli as a function of diameter. The smallest detectable particles V_Tmin at the four altitudes had terminal velocities between 0.83 and 0.87 m s⁻¹. We chose a single, conservative value of 0.8 m s⁻¹ to estimate the terminal velocities of the smallest particles detected by the profiler at all altitudes between the top of the bands and 1 km above the melting level, because the range of snowflake fall speeds is narrow (Atlas et al. 1973).

Figure 4a shows the geometry of the MIPS profiler vertical beam. Because of the 9° beamwidth, there is a contribution to the radial velocity by the mean horizontal wind that contributes to broadening of the power spectrum. The greatest contribution to the radial velocity comes from particles near the edge of the beam (Fig. 4b). Further broadening is induced if vertical shear of the horizontal wind is present, because the radial velocity of the particles near the edge of the beam will vary across a range gate. Last, turbulence induces variations in the vertical motion field, which further broadens the power spectrum.

To account for each of these factors, we took the following approach. The wind speed u and vertical shear of the horizontal wind u′ at each range gate were determined through temporal interpolation of three soundings (1200, 1800, and 0000 UTC 25–26 January 2004) taken at Lincoln, Illinois (KILX), for C1. For C2, these parameters were determined through temporal and spatial interpolation of the 1200 and 0000 UTC 15–16 February 2004 soundings from Nashville, Tennessee (KBNA), Peachtree City, Georgia (KFFC), and Shelby County Airport near Birmingham, Alabama (KBMX). The winds for C3 were determined from the same sounding locations but using the 1200, 0000, and 1200 UTC 25–26 February 2004 soundings. Select soundings are shown in section 3. Before using the rawinsonde-derived winds, we experimented with using the winds derived from the off-zenith (23°) beams of the profiler. Because of the low power returned from ice particles or clear air outside the bands, long averaging periods were required to determine the winds. However, these long time periods incorporated times for which the precipitation was not uniform across all beams. These inhomogeneities in the precipitation field at the levels of interest led to large uncertainties in the recovered winds and, as a result, derived vertical motions that were often physically implausible. For this reason, we chose to use the rawinsonde-derived winds. The uncertainties in the recovery of vertical motion using the rawinsonde winds are addressed below. Last, to estimate the turbulence we assumed that the vertical wind fluctuations can be scaled with the vertical shear of the horizontal wind (Rogers et al. 1996), because vertical wind shear will induce tumbling motions in air.

With these assumptions, we can write the maximum radial velocity V at the edge of the profiler beam associated with smallest detectable particles as

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where u is the horizontal wind at the center of each range gate; u′ = (∂u/∂z)(Δz/2), where ∂u/∂z is the wind shear and Δz is the gate spacing (105 m); θ is equal to 4.5°, the half-beamwidth of the profiler vertical beam; and w′ is the maximum variation in vertical velocity associated with turbulence, which is assumed to be equal to u′. Solving for w, we obtain

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Equation (6) reduces to Eq. (3) in Battan (1973) for the case of an infinitely small beamwidth in which the horizontal wind has no effect on w.

The uncertainty in the estimation of w is given by

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The root-mean-square (rms) estimate of the uncertainties ΔV, Δu, Δu′, ΔV_Tmin, Δw′, and Δθ were determined in the following manner. The variable u was determined from spatial and temporal interpolation of the wind speed u from soundings, assuming linearity. At worst, u could have had the value u of any of the soundings bounding the interpolation. The uncertainty Δu was therefore determined by considering the rms difference between the interpolated value of u and each of the measured u from all of the soundings used in the interpolation. The uncertainty Δu′ was determined in a similar way, except considering the sounding values of wind shear within the distance of a range gate. The uncertainty Δw′ was assumed to equal Δu′. The uncertainty ΔV was assumed to be the half-width of a spectral bin, 0.14 m s⁻¹. The uncertainty ΔV_Tmin based on the calculation of the terminal velocity of the smallest detectable particles was estimated to be 0.1 m s⁻¹. The uncertainty Δθ was assumed to be 1°. These uncertainties were determined for each individual band, because the sounding winds were different in each cyclone. Table 1 summarizes the values of the variables involved in the calculation of the uncertainty in vertical motion, and Δw for each of the nine bands, each cyclone, and the complete analyzed dataset. The uncertainty Δw ranged from 0.4 to 0.8 m s⁻¹, with the overall Δw for the entire dataset equal to 0.6 m s⁻¹.

a. C1, 25 January 2004, southern Illinois

The cyclone formed over Colorado around 0000 UTC 25 January 2004 and progressed southeastward into Oklahoma and Arkansas by 1200 UTC 25 January. Figures 5a,b and 6a,b depict the frontal structures associated with C1 at the 700-hPa level at 1200 UTC 25 January and 0000 UTC 26 January. The area sampled by the profiler was along a warm-frontal boundary that extended from Missouri through South Carolina at 1200 UTC and moved north of the Ohio River Valley by 0000 UTC. A region of high relative humidity was present along the warm front, which was coincident with the region of precipitation illustrated on the radar composite (Fig. 7a).

The precipitation began at the MIPS location in Flora at 1330 UTC as a series of bands passed over southern Illinois. The SNR profile during the precipitation event (1300–2200 UTC) revealed finescale bands that extended from the ground vertically to as high as 6 km (Fig. 8a). The bands ranged from approximately 5 to 35 km wide, were oriented roughly west to east, and moved northward at approximately 12 m s⁻¹. The melting level, apparent by relative maxima of SNR and a sharp gradient of radial velocity in Figs. 8a and 8b, respectively, was at 1.5 km for the first half of the precipitation event but rose to almost 2 km by 1900 UTC. The surface precipitation was in the form of freezing rain from 1330 through 1500 UTC and then turned to mixed rain and ice pellets. Between 1515 and 1545 UTC, a wide, elongated band passed over the profiler (band B). The precipitation at the surface during the passage of the band was entirely ice pellets. There was a lull in the precipitation around 1650 UTC, during which there was only a fine freezing drizzle, but light sleet resumed by 1700 UTC. The sleet alternated between light and heavy from 1730 through 1915 UTC before the precipitation stopped. Between 2000 and 2130 UTC, small drops of light freezing rain/drizzle mixed with small ice pellets fell intermittently. Approximately 3 mm (1/8 in.) of ice and 25 mm (1 in.) of sleet had accumulated during the 9-h period of precipitation at the profiler site in Flora.

A sounding launched at KILX at 1800 UTC during the passage of the bands over Lincoln (about 180 km northwest of the profiler) is representative of the band environment (Fig. 9a). The sounding showed an inversion up to 900 hPa, with a stable, saturated atmosphere through 450 hPa, the approximate top of the bands in Fig. 8a. By 0000 UTC 26 January, the precipitation had ended at the MIPS location.

b. C2, 15 February 2004, northern Alabama

This cyclone was associated with a weak surface low over Mississippi at 0000 UTC 15 February 2004. At this time at 700 hPa, a region of high relative humidity extended from Georgia westward into Arkansas (Figs. 5c and 6c). This region of moisture in which the bands formed was associated with a tongue of warm air—a feature termed the trough of warm air aloft, or trowal (see Martin 1998). The trowal progressively sharpened (Fig. 5d) during the period that the bands passed over the profiler in Huntsville (0300–1700 UTC 15 February, Fig. 8c).

Figures 9b and 9c show soundings taken at KBMX at 0000 UTC 15 February and KBNA at 1200 UTC 15 February. Both soundings were taken near the time that the profiled bands passed over these locations. At KBMX, a stable layer was present to the 850-hPa level, with a moist-adiabatic profile above that level. At KBNA, the atmosphere was dry and stable up to the 750-hPa level. Above that level, the temperature profile was moist adiabatic.

The precipitation started as rain at about 0300 UTC in Huntsville. The bands between 0600 and 1200 UTC were 20–50 km wide and extended up to 6 km (Figs. 7b and 8c). Figures 5d and 6d show that by 1200 UTC the coldest/driest air at 700 hPa was positioned directly south of the MIPS location. At this time, the trowal was apparent over southern Tennessee and northern Alabama in the θ field. The precipitation changed to mixed rain, snow, and drizzle near 1200 UTC and continued as mixed precipitation until it ended at 1800 UTC. The melting level was around 2.5 km from 0300 through 1000 UTC, gradually decreasing to 1.25 km by 1300 UTC (Figs. 8c,d).

c. C3, 25–26 February 2004, northern Alabama

At 1200 UTC 25 February 2004, a weak surface low pressure system was located over the Gulf of Mexico, south of Alabama. The low formed along a preexisting cold front that had moved southward across the eastern United States. At 700 hPa, a band of moisture stretched along and over this frontal boundary (not shown).

By 2100 UTC 25 February, a narrow region of precipitation extended in an arc from northeast Texas, northeastward to northern Alabama, and then southeastward into Georgia along this moisture axis (Fig. 7c). A second area of precipitation extended along the Alabama–Mississippi border. Both areas of precipitation passed over the profiler during the event.

The area of precipitation along the arc reached the profiler location in Huntsville around 1700 UTC 25 February. Several narrow bands (10–35 km in width) produced rain as they passed over the profiler. The bands were 4–4.5 km deep between 1700 UTC 25 February and 0000 UTC 26 February (Fig. 8e). The mean orientation of the bands was from west to east as they moved north-northeastward (Fig. 7c). The height of the melting level was around 1.75 km during this time period (Figs. 8 e,f).

By 0000 UTC 26 February, the surface low had just moved onshore over southern Alabama. A trowal was present at 700 hPa from southeastern Arkansas through western Georgia (Figs. 5e). Although the moisture boundaries were sharp (Fig. 6e), the thermal gradients in this cyclone were weak at 700 hPa at 0000 UTC (Fig. 5e). Figure 9d shows the KBMX sounding for 0000 UTC 26 February. At this time a stable layer was present between 900 and 750 hPa, with a moist-adiabatic temperature profile between 750 and 450 hPa (the approximate top of the bands). Saturated air was present from 950 to 850 hPa and from 700 to 550 hPa, with dry air above.

The area of precipitation that was along the Alabama–Mississippi border at 2100 UTC (Fig. 7c) reached the profiler location around 0100 UTC 26 February. At this time, the bands changed characteristics in that they increased to 6 km in height and were approximately 30–60 km wide. Shortly after 0400 UTC 26 February, the height of the melting level fell abruptly to just under 0.5 km (Figs. 8e,f). The precipitation changed from rain to a mixture of rain and wet snow until it ended after 0800 UTC.

The low pressure system had shifted eastward and weakened by 1200 UTC 26 February, although a weak trowal structure was still present but was located from central Tennessee through South Carolina (Fig. 5f). The KBMX sounding showed a saturated layer from the ground up to 725 hPa and was dry above, with a stable layer between 925 and 850 hPa (Fig. 9e).

a. Cyclone C1

The vertical velocity w within bands A–C of C1 is contoured in Figs. 10a–c. The portion of the bands for which w was determined is outlined and overlaid on the SNR fields (Figs. 10d–f). Figures 11 and 12 are arranged similarly for bands D–F in C2 and bands G–I in C3, respectively. All heights in these and subsequent profiler figures are above ground level (AGL). Ground level is approximately 140 and 200 m above sea level in Flora and Huntsville, respectively.

Band A crossed the profiler in 19 min, corresponding to an approximate width of 7 km based on the average cross-band wind component within the layer for which w was calculated, in this case between 2.5 and 5.0 km. Values of vertical motion ranged from −1.4 to 3.3 m s⁻¹ between 2.5 and 5.0 km (Fig. 10). Positive values of w were found throughout most of the band, with a mean updraft of 1.2 m s⁻¹ (standard deviation σ = 0.6 m s⁻¹; see Table 2) and the highest values (1.5–3.3 m s⁻¹) located between 2.5 and 3.5 km near the center. The regions of strongest positive vertical motion (updraft) generally corresponded to areas of high SNR (1.5–11 dB; Fig. 10d). Negative values of w averaged around −0.5 m s⁻¹ (σ = 0.3 m s⁻¹), were found along the edges of the analysis area, and generally corresponded to the lowest values of SNR (from −3 to −15 dB).

Band B had an approximate width of 31 km and took about 39 min to pass over the profiler. Values of w ranged between −2.2 and 3.6 m s⁻¹. The mean upward velocity was 1.6 m s⁻¹ (σ = 0.8 m s⁻¹). Figures 10b and 10e show that the strongest updrafts (2.0 ≤ w ≤ 3.6 m s⁻¹) were found within a widespread area near the center of the band between 2.5 and 4.5 km from 1530 to 1540 UTC, coinciding with the highest values of SNR (3–9 dB). Downdrafts (−2.2 ≤ w ≤ −0.5 m s⁻¹, average of −1.0 m s⁻¹, and σ = 0.5 m s⁻¹) were located along the edges of band B and were associated with regions of weak SNR (from −5 to −15 dB).

Band C passed over the profiler in a period of 41 min and was approximately 35 km wide. The range of vertical motion was from −1.9 to 2.9 m s⁻¹ between 2.5 and 5.5 km (Fig. 10c). The average upward velocity was 1.0 m s⁻¹ (σ = 0.7 m s⁻¹), with the strongest updrafts (1.5 ≤ w ≤ 2.9 m s⁻¹) between 2.5 and 3.5 km through the width of the band. Again, the regions of updrafts were coupled with regions of high SNR (2–10 dB) while the regions of stronger downdrafts (w ≤ −0.5 m s⁻¹, average of −0.8 m s⁻¹, and σ = 0.4 m s⁻¹) were coupled with areas of weak SNR (from −3 to −15 dB; Fig. 10f) along the outer edges of the band.

In qualitative terms, a correlation between the values of SNR and vertical motion in Figs. 10a–f is evident. Figures 13a–d show a quantitative analysis of the correlation for bands A, B, and C and the overall correlation for C1. Bands A and C had the highest correlation coefficients r of 0.85, and band B had the lowest (r = 0.76). The correlation coefficient between all values of SNR and w within C1 was 0.79.

Figures 14a–d show cumulative frequency diagrams for w and the uncertainty in w (Δw) for each band within C1 and the overall analysis for C1. These diagrams show the percent of the total number of calculated values less than w. For example, Fig. 14a shows that 32% of the 83 samples for band A were negative (downdrafts), 43% were updrafts exceeding 1 m s⁻¹, and 7% exceeded 2.0 m s⁻¹. Only 14% of the 234 observations were negative in band B, 68% exceeded 1 m s⁻¹, and 25% exceeded 2.0 m s⁻¹ (Fig. 14b). In band C, 16% of the 197 observations were negative, 37% exceeded 1 m s⁻¹, and 9% exceeded 2.0 m s⁻¹ (Fig. 14c). Figure 14d shows that 17% of the 514 total observations for C1 were negative, over one-half (52%) were greater than 1 m s⁻¹, and 16% exceeded 2 m s⁻¹. The mean updraft (downdraft) velocity for C1 was 1.3 m s⁻¹ (−0.8 m s⁻¹), with σ = 0.8 m s⁻¹ (0.5 m s⁻¹). Table 2 summarizes the statistics for the analysis of the vertical motion in this cyclone, as well as for C2 and C3.

b. Cyclone C2

Band D, a combination of two narrow bands with little separation between them, had a total width of ∼52 km and took 45 min to pass over the profiler. Values of w deduced between 3.5 and 6.0 km ranged from −4.3 to 4.3 m s⁻¹ (Fig. 11a). The SNR field in Fig. 11d suggests that the two bands were sloped and that the tilted regions of strongest SNR (5–10 dB) corresponded to the regions of strongest updrafts (1.5 ≤ w ≤ 2.9 m s⁻¹; Fig. 11a). A small area of negative w (from 0 to −0.7 m s⁻¹) was located between the two narrow bands. Another region of downward motion (from 0 to −2 m s⁻¹) was located between 3.5 and 4.0 km from 0552 to 0605 UTC, coincident with the underside of the first tilted band. Negative values of w (from 0 to −4.3 m s⁻¹) were also found along the edges of the bands, consistent with the areas of weak SNR (from −4 to −15 dB). The mean positive (negative) w was 1.2 m s⁻¹, with σ = 0.8 m s⁻¹ (−1.1 m s⁻¹, with σ = 1.0 m s⁻¹).

Band E was approximately 33 km wide and took 28 min to cross the profiler (Fig. 11b). This band had the strongest updraft in the nine bands analyzed. The strongest updraft (6.7 m s⁻¹) took place in the center of the band between 0852 and 0856 UTC and corresponded directly with a small area of high SNR (10–17 dB; Fig. 11e). Values of w derived between 3.5 and 6.0 km ranged from −2.5 to 6.7 m s⁻¹. The mean updraft speed was 1.5 m s⁻¹ (σ = 1.6 m s⁻¹) throughout the band, and the mean downdraft was −0.7 m s⁻¹ (σ = 0.7 m s⁻¹). Regions of negative w (to −2 m s⁻¹) were found on the underside of the strong-echo region extending to the left side of the figure between 4 and 6 km from 0840 to 0850 UTC (Fig. 11b), with regions of weak upward motion (up to 1.5 m s⁻¹) within the echo region. In addition, downward motion was found along the edges of the band and also between two peaks in SNR and positive w between 0857 and 0907 UTC. Downdrafts correlated well with the regions of weak SNR (from −6 to −12 dB; Figs. 11b, e). Note that band E is unique in this dataset in that it can be classified as being convective based on the absence of a bright band in the SNR field, whereas the other bands appear to be more stratiform.

Band F consisted of two subbands that were only separated above 3.5 km (∼1 km above the melting level, Fig. 11f). Band F passed over the profiler in 51 min, corresponding to a distance of about 58 km. Vertical motion between 3.5 and 6.0 km ranged from −2.3 to 1.9 m s⁻¹. The mean upward motion within band F was 0.8 m s⁻¹ (σ = 0.4 m s⁻¹). Two regions of maximum updrafts (1.0–1.5 m s⁻¹) were present between 3.5 and 4.0 km near the center of each subband that corresponded to two regions of high SNR (2–7 dB). Downdrafts (from 0 to −2.3 m s⁻¹) were found along the edges of the band and were consistent with low values of SNR (from −5 to −15 dB; Figs. 11c, f). The mean negative w was −1.0 m s⁻¹ (σ = 0.6 m s⁻¹).

Although there is an obvious correlation between SNR and w that is apparent in Figs. 11a–f, the correlation was not as strong as that in C1 (r_C1 = 0.79). The overall correlation coefficient in C2 for all of the bands was 0.59 (Fig. 13h). Band D had the lowest correlation coefficient of 0.60, and band F had the highest at 0.71 (Figs. 13e–g).

Figures 14e–h show cumulative frequency diagrams of w and Δw for bands D–F and the overall analysis for C2. In band D, 35% of the 159 observations were negative, 39% exceeded 1 m s⁻¹, and 8% were greater than 2 m s⁻¹ (Fig. 14e). Band E had a wide range of values of w, with 29% being negative, 34% being greater than 1 m s⁻¹, 19% being greater than 2 m s⁻¹, and 5% exceeding 5 m s⁻¹ (Fig. 14f). In a similar way, 28% of the values of w were negative in band F; however, there were no updrafts greater than 2 m s⁻¹, although 27% exceeded 1 m s⁻¹ (Fig. 14g). Figure 14h shows that 31% of the 421 total observations during C2 were negative, 34% exceeded 1 m s⁻¹, and 9% exceeded 2.0 m s⁻¹. The mean positive updraft velocity for C2 was 1.1 m s⁻¹ (σ = 1.1 m s⁻¹), and the mean downdraft velocity was −1.0 m s⁻¹ (σ = 0.8 m s⁻¹).

c. Cyclone C3

Band G took 40 min to pass over the profiler and had an approximate width of 18 km. Values of w ranged between −1.0 and 2.7 m s⁻¹ between 3.0 and 4.6 km (Fig. 12a), with an average updraft speed of 0.6 m s⁻¹ (σ = 0.5 m s⁻¹). The strongest updraft (1.0–2.7 m s⁻¹) was located between 3.0 and 3.5 km from 2100 and 2105 UTC, toward the left side of the band (Fig. 12a), and matched up with the higher values of SNR (4–14 dB; Fig. 12d). This band contained a large area of negative w values (average of −0.3 m s⁻¹, with σ = 0.2 m s⁻¹), which correlated well with weak values of SNR (from −4 to −15 dB; Fig. 12d).

Band H took 48 min to pass over the profiler and had an approximate width of 38 km. The vertical motion ranged from −2.1 to 2.4 m s⁻¹ between 3.0 and 6.0 km (Fig. 12b), with a mean updraft (downdraft) speed of 0.9 m s⁻¹, with σ = 0.5 m s⁻¹ (−0.7 m s⁻¹, with σ = 0.5 m s⁻¹). There were two areas of maximum w (1.0–2.0 m s⁻¹) and corresponding maximum SNR (2–12 dB; Figs. 12 b,e); the first occurred from 0050 to 0100 UTC between 3.0 and 3.5 km toward the beginning of the band (left side of Fig. 12b), and the second was from 0110 to 0125 UTC and stretched up to 5.0 km in the last two-thirds of the band (right side of Fig. 12b). The downdrafts (−2.1 ≤ w ≤ 0 m s⁻¹) were mainly confined to the edge of the band and corresponded to regions of weak SNR (from −7 to −15 dB; Fig. 12e).

Figure 12c shows that band I took 50 min to pass and corresponded to a width of approximately 37 km. Its vertical velocities ranged between −1.9 and 3.0 m s⁻¹ from 3.0 to 6.0 km, with the average positive w equal to 0.8 m s⁻¹ (σ = 0.5 m s⁻¹). The maximum values of w (1.0–3.0 m s⁻¹) occurred between 3.0 and 4.0 km through the width of the band and were associated with regions of high SNR (1–15 dB), which is apparent in Fig. 12f. Downdrafts (−1.9 ≤ w ≤ 0 m s⁻¹, average of −0.7 m s⁻¹, and σ = 0.4 m s⁻¹) were located along the top edge of the band and correlated well with lower values of SNR (from −5 to −14 dB, Fig. 12f). Bands G–I in C3 all had very high correlations between SNR and w (Figs. 13i–l). The overall correlation coefficient for C3 was 0.76, with band I having the highest value (r = 0.85) and band G having the lowest (r = 0.68).

The cumulative frequency diagram of w and Δw in Fig. 14i shows that 55% of the w values were negative in band G, 9% exceeded 1 m s⁻¹, and only 1% exceeded 2 m s⁻¹. In band H, 27% of the values of w were negative, 32% exceeded 1 m s⁻¹, and only 2% were greater than 2 m s⁻¹ (Fig. 14j). In band I, 38% of values of w were below 0 m s⁻¹, 19% exceeded 1 m s⁻¹, and 3% were greater than 2 m s⁻¹ (Fig. 14k). With the three bands combined in Fig. 14l, 37% of the 580 total derived values of w were negative, 22% exceeded 1 m s⁻¹, and 2% were greater than 2 m s⁻¹. The mean positive vertical motion for C3 was 0.9 m s⁻¹ (σ = 0.5 m s⁻¹), and the mean downward motion was −0.6 m s⁻¹ (σ = 0.4 m s⁻¹).

The correlation coefficient between the SNR and w for all nine bands in the three cyclones was 0.68 (Fig. 15). The cumulative frequency diagram for the overall analysis showed that 29% of the 1515 total derived values of w were negative, 35% exceeded 1 m s⁻¹, and 9% exceeded 2.0 m s⁻¹ (Fig. 16). The minimum derived value of w was −4.3 m s⁻¹ (in band D) and the maximum value was 6.7 m s⁻¹ (in band E). The mean value of w for all cyclones was 0.6 m s⁻¹, with a standard deviation of 1.1 m s⁻¹, the mean upward vertical motion was 1.1 m s⁻¹ (σ = 0.9 m s⁻¹), and the mean downward motion was −0.7 m s⁻¹ (σ = 0.6 m s⁻¹). Table 2 summarizes these observations for all of the nine bands studied.