a. Coherent air scattering

The theory of scattering from turbulent fluctuations in the refractive index of the air is well developed. The radar cross section per unit volume η (radar reflectivity) is given by (Ottersten 1969):

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where φ_n(k) is the space spectrum, that is, the Fourier transform of the three-dimensional spatial covariance function of refractive index fluctuations. The vector k is the wavenumber vector in the radar radial direction. Its absolute value is related to the radar wavelength λ through k = 4π/λ. Equation (1) expresses that the radar sees a narrow continuum of fluctuations on spatial scales around λ/2. (Not λ, but λ/2, because backscattering is considered.) When the fluctuations are distributed isotropically in space, φ_n(k) does not depend on the direction of k. For homogeneous isotropic turbulence there exists an inertial subrange in which φ_n(k) follows a power law (Tatarski 1961):

φ_nkC²_nk^−11/3(2)

where C²_n is the structure parameter of the refractive index fluctuations and is a measure of the strength of the refractive index variations at all scales. Ottersten (1969) points out some other useful relations for isotropic turbulence, for example, for the refractive index variance spectrum:

E_nkπk²φ_nk(3)

E_n(k) follows the famous “−5/3 law” of Kolmogorov in the inertial subrange. Combining Eqs. (1) and (2) and substituting k = 4π/λ leads to the well-known expression

ηλC²_nλ^−1/3(4)

The coherent air scattering from refractive index fluctuations thus has a weak wavelength dependence in the inertial subrange. Variations in water vapor contribute most to the clear-air scattering in the lower atmosphere (Gossard and Strauch 1981).

Equation (4) is valid for radar wavelengths larger than a critical wavelength λ_c, which is defined as 8πη₀, where η₀ is the Kolmogorov microscale (Gossard 1984a). The Kolmogorov microscale varies over the range 0.4–1.25 mm in cumulus clouds (Shaw et al. 1998), so that λ_c varies over the range 1–3 cm. Under conditions of weaker turbulence λ_c can sometimes become larger than 10 cm (Gossard et al. 1984b; Gage et al. 1999). The X-band radar wavelength will be sensitive to turbulence in cumulus clouds most of the time, but under conditions of weaker turbulence even S-band radars may measure outside the inertial subrange.

b. Particle scattering

If the wave transmitted by a radar goes through a volume containing particles, part of the energy is scattered back to the radar. The waves scattered back by the particles interfere with each other. If the particles are randomly spaced, then some configurations of particle positions will show a net constructive interference, while other configurations have a net destructive interference, but, on average, the interference is zero. The received power, when averaged over all possible configurations of the particle positions, is equal to the sum of the powers scattered back by the individual particles. There is no net interference and one speaks of “incoherent scattering.” The radar reflectivity for incoherent scattering from a volume V with N_v small spherical particles is given by (Battan 1973):

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This expression is valid for particles with diameters D_i much smaller than the wavelength λ. K = (m² − 1)/(m² + 1), where m is the complex index of refraction of the material of the particle (|K|² equals 0.93 for water and a wavelength of 10 cm).

When the particles are not completely randomly positioned, a net interference may result and the radar reflectivity differs from Eq. (5). Several scattering theories are known in the literature that take interference into account. Two of these will be briefly reviewed below and compared with each other. They mainly differ in their way of modeling the coherent scattering.

Siegert and Goldstein (1951) treat the backscattering from assemblies of scatterers. For simplicity they assume that all the scattering particles return a signal of equal magnitude. Defining N(r)dr as the number of scatterers between a distance r and r + dr from the radar, they derive the following formula for the cross section σ:

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In this equation, p is a proportionality constant that is assumed to be the same for all scatterers, k equals 4π/λ, and the overbar denotes a time average. The first term is the incoherent contribution to the cross section. It results from the assumption that the scatterers are independent, at least in the sense that the presence of a certain scatterer in any interval does not prejudice the presence of other scatterers. The second term in Eq. (6) is called the coherent term. It expresses the coherent contribution to the cross section due to deterministic spatial variations in the concentration. For example, the backscattering due to large-scale gradients is included in this term. This term equals zero if the time-averaged concentration has no fluctuation on a spatial scale of half the wavelength. The term could be important when, for example, there are sudden spatial changes in refractive index of the atmosphere. The treatment of Siegert and Goldstein is incomplete in the sense that Eq. (6) does not include the coherent scattering from spatial fluctuations caused by, for example, turbulent mixing. Turbulent mixing can cause the scatterer concentration to vary both in space and time. This means that the time average of the concentration can be a constant, that is, independent of position, but still at any particular time there can be a spatial correlation in the concentration, just as there can be a correlation in the values that the concentration has in time, at any given position. The spatial correlation can cause a coherent return. This fact is not included in Eq. (6).

Gossard and Strauch (1983) do take spatial correlations in concentration into account. Let N(r)dr be the number of drops in the position increment between r and r + dr. The radar backscattering is proportional to a factor I, which is given by

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The overbar denotes a time average. For backscattering, the wavenumber vector k has an absolute value equal to 4π/λ and is directed from the scatterers to the radar. It is again assumed for simplicity that all the scatterers have the same cross section. Gossard and Strauch split N(r) into two terms, a deterministic and a random part:N = N(r) + δN(r). The deterministic part gives a contribution identical to the second term on the right-hand side of Eq. (6) and will here be further ignored. The random part has a time average that is zero and gives the contribution

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If a spatial correlation function is defined as follows:

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where V is the scattering volume, then Eq. (8) can be expressed as

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The integral in Eq. (10) is a Fourier integral and I can be written in terms of the three-dimensional spectrum φ_N(k) of concentration fluctuations as follows:

Iπ³Vφ_Nk(11)

Gossard and Strauch argue that if there is no spatial correlation between neighboring parcels, the backscattered power is proportional to the total number of scatterers in the scattering volume; the usual incoherent return. This is due to the fact that the number of scatterers in a parcel has a Poisson distribution and this distribution has a variance equal to the mean. The incoherent scattering is the same as the first term in Eq. (6). If there is a correlation between the concentrations in parcels that are a certain distance apart, then there will be a coherent contribution to the scattering expressed by Eq. (11), which we call the “stochastically coherent” contribution. Its strength depends on the amount of fluctuations on scales close to half a wavelength.

Gossard and Strauch do not mention that the incoherent term is always present, even when there is correlation between concentrations in neighboring parcels (Erkelens et al. 1999a). This is caused by the fact that the scatterers do not form a continuum but are discrete points in space. The scatterer concentration is a function of space and time. Neglecting any large-scale gradients, there is a global mean of the concentration for the entire scattering volume. A process such as turbulent mixing can cause fluctuations in the concentration around the global mean. These fluctuations vary in space and time and show correlation between different parcels. Even when the total number of scatterers in a certain parcel is correlated with the total number in a parcel nearby, the exact position of each individual scatterer in the parcel is random (unless the mutual distances between the particles are completely specified, but that is a deterministic case). Therefore, the number of scatterers in small parcels with dimensions of the order of the average distance between the scatterers or smaller has a Poisson distribution with an (ensemble) variance equal to the local expected number of scatterers in the parcels. This local expected value is determined by the physical process that creates the correlations between the number of particles in different parcels. In other words, the turbulence or some other physical process causes correlations between the particle concentrations of the parcels, but on top of that we have the Poisson statistics associated with the random nature of the individual scatterer positions. [This means that the total variance of the number of scatterers in a certain volume is larger than one would expect for completely randomly distributed scatterers, that is, the scatterer distribution exhibits super-Poissonian variance. Kostinski and Jameson (2000) introduced super-Poissonian variance to describe variations in clouds.]

The incoherent backscattering, due to the random positions of the individual particles, and the coherent backscattering, due to the spatial correlations in locally mean scatterer concentration, are additive. Equation (11) describes the coherent backscattering for which the theory for scalar conservative passive additives in homogeneous, isotropic turbulence predicts a k^−11/3 dependence (Tatarski 1961). However, in the theory the concentration is considered as a continuous function in space and time. As explained above, the discrete nature of the particles is the cause why the usual incoherent scattering is also present, in addition to the coherent scattering.

c. Radar reflectivity factor

In this section an expression is derived for the total radar reflectivity factor for the case where fluctuations are caused by turbulent mixing and the particles can be considered as “conservative passive additives.” The presence of scatterers changes the refractive index of a vacuum. If the particles are small compared with the wavelength, then the refractive index n of the medium as a whole is approximated by (see, e.g., Van de Hulst 1981, p. 67):

nπαN,(12)

where N is the concentration of scatterers and α the polarizability of the scatterers. For small spheres, the polarizability is given by (Van de Hulst 1981, p. 70):

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where ε_r is the permittivity of the material of the scatterer. The variance of the refractive index due to variations in scatterer mass concentration is given by

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To find an expression for the radar reflectivity for the coherent scattering, varn has to be related to C²_n and an expression for var(ND³) is needed. For this, we make use of the fact that the three-dimensional spectrum of isotropic refractive index fluctuations is normalized such that (Ottersten 1969):

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If it is assumed that: 1) there are no variations on scales larger than L₀, the outer scale of the inertial subrange, and 2) the inner scale is much smaller than the outer scale, then, putting Eq. (2) into Eq. (15) and integrating from 2π/L₀ to infinity, leads to (Gage and Balsley 1980):

C²_nL^−2/3₀n,(16)

Because we want to derive an expression for the coherent contribution to the radar reflectivity factor, fluctuations in n due to the random positions of the individual particles are ignored for now. As explained in section 2b, these fluctuations give rise to the usual incoherent scattering term.

The coherent backscattering depends on the square of the concentration. The incoherent radar return depends linearly on the concentration. Both the incoherent and coherent returns depend on the diameter to the power of six. The coherent term in Eq. (17) becomes increasingly important for longer wavelengths and higher particle densities. If we consider the following range of values of the parameters: β: 0.05–1, L₀: 10–500 m, and N: 100–1000 cm⁻³, then the ratio of the coherent to the incoherent term of Eq. (17) spans the following range:for a wavelength of 3 cm the ratio lies between −44 dB and +4 dB, for 10 cm between −24 dB and +23 dB, and for 30 cm between −7 dB and 40 dB. Hence, coherent particle scattering might in some cases be significant at X-band, it can very well be important at S-band, and even more so for UHF radars.

Equation (17) is applicable when the fluctuations in particle concentration are caused by mixing between air volumes having large differences in concentration. Several recent studies show that fluctuations in particle concentration can appear in turbulent flows due to the inertia of the particles, without the need for mixing with air with a different mean concentration (see, e.g., Squires and Eaton 1991; Vaillancourt 1998; Shaw et al. 1998). The particles are preferentially concentrated by coherent turbulent structures. The importance of the effect in clouds is, however, still under debate (see, e.g., Grabowski and Vaillancourt 1999; Shaw et al. 1999). The fluctuations that are found with this preferential concentration mechanism generally appear on scales of a few centimeters. The mechanism works for a limited range of particle masses: very light particles just follow the flow, while for heavier particles gravitational sedimentation becomes important. If the particles are too heavy, they just fall through the turbulent structures and there is not enough interaction. The total radar reflectivity factor for a volume where the preferential concentration mechanism is operating will also have an incoherent component ND⁶ and a coherent component proportional to (N)²D⁶, but the wavelength dependence and the strength of the coherent component could differ much from Eq. (17).

d. Dual-frequency measurements

If dual-frequency measurements of purely coherent scattering are compared, a difference between the measured radar reflectivities will be found. If both wavelengths fall within the inertial subrange, this difference can be calculated from

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where Z₁ and Z₂ are the reflectivity factors for the two wavelengths, expressed in dBZ, and log is the logarithm of base 10. For example, in sections 3 and 4 dual-wavelength measurements of developing cumulus clouds and smoke are discussed. The wavelengths used are 3 and 10 cm for the clouds and 3.2 and 33 cm for the smoke. Equation (18) predicts differences of 19 and 37 dBZ for purely coherent scattering for these pairs of wavelengths. Dual-frequency measurements of purely incoherent scattering by small spheres should by definition give a difference of 0 dB. If both incoherent and coherent scattering contribute to the return, the measured difference should lie somewhere between 0 dBZ and the value predicted by Eq. (18). This happens, for example, when the radar return at the longer wavelength is dominated by coherent scattering from humidity and temperature variations and the return at the shorter wavelength by incoherent scattering from particles. Values between the limits are observed for the majority of the measurements of developing cumulus clouds (Knight and Miller 1998), and for smoke (Rogers and Brown 1997). The puzzling aspect of these measurements is that both sets of measurements show a correlation between the reflectivity factors at two wavelengths, which one does not expect when the origins of the radar returns are different. The measurements will be discussed further in the following sections.

a. Description of the measurements

In a recent article, Knight and Miller (1998) discuss measurements of developing cumulus clouds, performed with two radars: an X-band radar with a wavelength of 3 cm and an S-band radar with a wavelength of 10 cm. Two examples are shown in Figs. 1 and 2. The measurements were obtained during the Convection and Precipitation Electrification Experiment (CaPE) and the Small Cumulus Microphysics Study campaigns in Florida in July and August 1991 and 1995. In these figures, DZS, DZX, and DZS−X denote the radar reflectivity factor at S band, at X band, and their difference, respectively. Most measurements could be explained by the traditional theory. For example: Coherent air scattering at the borders of the cumulus clouds is due to mixing of the humid cloudy air with dryer environmental air and can cause “mantle echoes” with differences in the reflectivity factors close to 19 dBZ, the value expected for coherent scattering at these wavelengths; or, when drizzle or rain droplets are present, the Rayleigh scattering from them can dominate, causing equal reflectivity factors. In these cases there is a correlation in the returns, as the scattering process dominating both wavelengths is the same. Also cases were observed in which the 10-cm radar measured coherent scattering, while the 3-cm radar measured incoherent scattering, and there was no correlation between the reflections.

However, often there was a correlation in the X- and S-band reflectivity factors, but their difference was not equal to 0 dBZ (the value for purely incoherent scattering) or 19 dBZ (for purely coherent scattering). For example, in Fig. 3 some of the X- and S-band reflectivity factors lie on a line with a slope of one with a difference of about 10 dBZ (but other offsets have been observed as well). No simple explanation for this can be found from the conventional theories of incoherent droplet scattering and coherent air scattering. The phenomenon was especially noticeable on days where the air outside the cloud was very humid, because the effect was not obscured then by coherent scattering due to water vapor fluctuations at the borders of the cloud. In section 3b, a possible explanation for this correlation will be presented in terms of coherent scattering from the droplets.

At the cloud bases of the developing cumulus clouds, Knight and Miller reported two important features: flat echo bases and weak echo regions. Flat echo bases are occasionally observed; they are very flat and horizontal bottoms in the X-band echo contours, similar to the visually flat cloud base at the condensation level. Figure 1 shows a clear example. Here the S-band reflectivity pattern also shows a flat echo base, but this is not always the case. Knight and Miller interpret the flat echo bases as the result of unmixed adiabatic ascent.

The base of the cloud in Fig. 1 is probably somewhere between 400 and 700 m. Near the cloud base, the S-band measurement shows a region with lower reflectivities. This region is called the “weak echo region” and is seen in nearly all the actively growing clouds. The bottom frame shows that the S-band reflectivity is dominated by coherent scattering, so the weak echo region is a minimum in the coherent return at S band. It is also sometimes present at X band but is then much less pronounced. The weak echo region at S band extends almost to the ground here. The weak echo regions often have reflectivity factors in the order of −20 dBZ.

Another feature reported by Knight and Miller is that on some days it is fairly common that the difference in reflectivity factors in the mantle echoes is not 19 dBZ but several decibels higher. This indicates a slope steeper than −5/3 for the variance spectrum of the refractive index variations. It may be that the X-band radar measured in the dissipation range, but a steeper slope may also be due to the effect of evaporation and condensation on the slope of the variance spectrum of humidity variations, as will be explained in section 3c.

b. Coherent particle scattering in cumulus clouds

In situ aircraft measurements of the droplet concentrations performed during CaPE in clouds very similar to the ones described above showed droplet concentrations reaching 800–900 cm⁻³ near cloud base (Paluch et al. 1996). There is much uncertainty about the values of β and L₀, and they may be related. A value of 10 m for L₀ was used by VanZandt et al. (1978) for the free atmosphere, but this value is probably much too small for the developing cumulus clouds considered here. Davis et al. (1996) measured the spatial fluctuations of liquid water content in marine stratocumulus clouds using a King hot-wire probe and Davis et al. (1999) used a particulate volume monitor for that purpose in broken stratocumulus decks with embedded towering cumulus. By dividing the standard deviation of the liquid water content measurements by their mean value, an estimate of the relative standard deviation β is obtained. This yields values between 0.05 and 0.58. A difference in the reflectivity factors of about 10 dBZ has been observed. Such a difference would result, for example, for L₀ = 150 m and β = 0.6. This result shows that coherent droplet scattering can be significant in S-band measurements of cumulus clouds. In Fig. 2, the strongest correlation between the radar reflectivity factors at X band and at S band occurs for values of the X-band reflectivity factor that lie between about −5 and 2 dBZ. For a concentration of 800 cm⁻³, this would mean droplet diameters between 27 and 35 μm (when the droplet size distribution is assumed to be monodisperse).

Since both the incoherent term and the coherent term in Eq. (19) depend on D⁶, we could explain the observed correlation by assuming that the differences between the measured data points are mainly due to an increase in the average diameter of the particles with height: the ascending air becomes colder and the droplets grow because of condensation.

Fluctuations on scales of about 5 mm to 5 cm have been found experimentally from FSSP data (Baker 1992) in cumulus cloud regions, which were picked by the eye to be homogeneous on large scales. Such centimeter-scale fluctuations might be caused by the preferential concentration mechanism mentioned earlier. The coherent scattering from such fluctuations can already be significant for very small values of the relative standard deviation β, if the fluctuations appear only on the small scales and not on larger scales. A very rough estimate can be obtained by assuming that the −5/3 law is valid for these centimeter-scale fluctuations. Then one may set L₀ in Eq. (19) equal to 5 cm to compute a coherent contribution that is about 10 dB stronger than the incoherent contribution for β as small as 0.04.

c. Coherent air scattering in clouds

In addition to the incoherent and coherent contributions to the radar reflectivity from particles, the air also supplies a coherent contribution. The refractive index of clear (i.e., particle-free) air depends on pressure, temperature, and humidity. Fluctuations in water vapor concentration dominate the scattering by clear air in the lower atmosphere (Gossard and Strauch 1981). Gossard (1979) has compared the relative magnitudes of coherent scattering from water vapor fluctuations in clouds with that of coherent scattering from droplets. He derives for the variations in refractive indices n_υ (due to vapor variations) and n_l (due to liquid variations):

View Expanded

respectively.

Here Q_υ and Q_l are the water vapor and liquid water mixing ratios (grams per kilogram of dry air) and “var” designates the spatial variance. Gossard and Strauch (1983) state that within a cloud in steady state at saturation with no precipitation removing water from the cloud and minimal entrainment, it seems reasonable to assume that varQ_υ is equal to varQ_l. Since the coherent scattering is proportional to varn, the contribution to coherent scattering from water vapor fluctuations is about 30 times stronger than coherent scattering from droplets for equal variations in water vapor content and liquid water content [see Eqs. (4) and (16)].

There are several mechanisms that influence the strength of the water vapor and liquid water fluctuations (Erkelens et al. 1999b) and it will be argued below that, in clouds, the variations in water vapor can be much smaller than those in liquid water. Consequently, in some cases coherent scattering from the cloud droplets can dominate coherent scattering from water vapor, contrary to common belief.

At the top and sides of convective clouds the cloudy air mixes with environmental air, leading to mantle echoes on the reflectivity patterns for longer wavelengths (Knight and Miller 1998). The environmental air does not contain water droplets, so mixing can cause large fluctuations in liquid water content. If, in addition, the air outside the cloud is very humid, the resulting fluctuations in the water vapor content can be smaller than those in the liquid water content. Environmental air may penetrate deeply into the cloud, indicated by the thickness of mantle echoes on dry days. The mantle echoes can sometimes be as thick as 1 km at the top of cumulus clouds.

Condensation and evaporation also influence fluctuations in the water vapor content and liquid water content. Condensation and evaporation will always tend to bring the relative humidity toward about 100% and are more effective for larger deviations from saturation. So this will reduce fluctuations in water vapor content but not necesarily the fluctuations in liquid water content. Rogers et al. (1994) report a case where the clear-air echo of a turbulent layer intensified when light rain fell through the layer. They analyze in detail the effect of evaporation of rain on C²_n and conclude that the thermodynamic effect of the rain on the layer is to reduce the reflectivity, contrary to what is observed. We wonder whether a coherent scattering contribution from the smallest rain droplets may cause the observed intensification.

The wavelength dependency of λ^−1/3 in Eq. (4) is valid for any conservative passive additive (CPA) in homogeneous isotropic turbulence in equilibrium (Tatarski 1961). For a CPA, the theory says that the fluctuations are generated at large scales and are passed on completely to smaller scales. Only at the smallest scales are they removed (by molecular diffusion). An interesting effect of condensation and evaporation is that it may affect the slope of the spatial spectrum of water vapor and liquid water fluctuations. (Water vapor content and liquid water content are not CPAs and the “−5/3 law” may not be valid.) Condensation and evaporation tend to reduce fluctuations in water vapor content. (This will happen at all scales, but more strongly at larger scales, because the largest deviations from saturation appear at the largest scales.) Due to this sink, not all of the fluctuations will be passed on to smaller scales. This means that the slope of the water vapor spatial spectrum could be steeper than that of a CPA.

An updraft may also reduce humidity fluctuations. If there is a mean upward motion of the air then the relative humidity in initially droplet free air will increase until saturation is reached. Then new droplets may form due to condensation. The cloudy air will remain at saturation, but the droplets will grow due to condensation. So, the effect of an updraft would be to reduce fluctuations in water vapor and at the same time increase fluctuations in liquid water, until saturation is reached in the clear air. This may lead to large fluctuations in liquid water concentration and slopes of the spatial variance spectra that differ from −5/3.

It is interesting to note that changes in the slope of the variance spectra can also occur when chemical reactions take place (Corrsin 1961; Pao 1964). The methods used in these references to compute the changes in the spectra might be applied to the clouds, but that is outside the scope of this paper.

There are also experimental indications of a deviation from the −5/3 law. Indications of a flatter slope have been found in a forward scattering experiment performed by Gossard and Strauch (1981). The dual-wavelength measurements of Gage et al. (1999) also show signs of a flatter slope. Davis et al. (1999) performed measurements of the liquid water spatial variance spectrum with a particulate volume monitor in stratocumulus clouds. The spectrum shows a slope that is less steep than −5/3 for scales smaller than a few meters.

Diffusion and sedimentation influence fluctuations in water vapor content or liquid water content as well. Diffusion reduces small-scale fluctuations in water vapor content, but this mechanism is negligible for water droplets. However, fluctuations in liquid water content may be removed by droplet sedimentation. The broader the droplet size distribution is, the larger the differences in fall velocities of the droplets, and the more effective sedimentation is in reducing variations in liquid water content.

All the mentioned processes can influence the strength of the fluctuations in water vapor content and liquid water content. A quantitative estimate of the effects is difficult because each of the processes involved has its own timescale. Estimates for some of the timescales involved can be found in, for example, Grabowski (1993).

Due to the effects discussed above, fluctuations in liquid water content may exist for longer times than those in water vapor content and may be much stronger. They may therefore penetrate the cloud more deeply. Mixing could be effective over larger parts of the clouds than might be concluded on the basis of the thickness of the mantle echo. For, the mantle echo mainly represents the area where fluctuations in water vapor content contribute strongly to the S-band reflectivity. Its apparent thickness, however, depends on the contrast in humidity between the saturated cloudy air and the subsaturated environmental air, on how fast humidity variations are reduced, and on the strength of the incoherent contribution.

Simultaneous measurements on centimeter scales of the water vapor and liquid water concentrations are needed to prove that the fluctuations in water vapor are really much smaller than those in liquid water. For a complete description, the covariances between humidity, temperature, and liquid water content should be measured also.

Some experimental evidence exists on the possibility of significant coherent particle scattering. Politovich and Cooper (1988) performed measurements of supersaturation in 147 cloud regions. The average supersaturation was near 0%, with relative standard deviations of 0.4% in 80% entrained air and 0.1% in the core of cumulus clouds. The regions giving a high correlation between S- and X-band reflectivities in the article by Knight and Miller (1998) are about 2 km above cloud base. An adiabatic cloud with a temperature of 25° Celsius and a pressure of 930 mb at cloud base will have a water vapor content of about 17 g m⁻³ and a liquid water content of about 6 g m⁻³ at 2 km above cloud base. Equations (4), (16), and (20) show that if coherent particle scatter is to dominate coherent air scatter the spatial standard deviations of the variations in liquid water content should be (58.5/2.09) = 5.3 times as large as those in water vapor content. The measurements of Politovich and Cooper would mean that the relative variations in liquid water content should dominate the scattering when they are above 6% for the entrained region and about 1.5% in the core of cumulus clouds. (Lower in the cloud the ratio of water vapor content to liquid water content will be higher, thus coherent air scatter will be more important near the cloud base.) Davis et al. (1996, 1999) found relative liquid water variations in stratocumulus clouds between 5% and 58% of the mean value. It is therefore very well possible that the coherent droplet scattering is stronger than the coherent air scatter in clouds, except close to the cloud base. Venema et al. (1999) have analyzed in more detail for which radar frequencies, how often, and in which types of clouds coherent particle scattering may be important.

a. Description of the measurements

Another measurement showing a puzzling correlation between the radar measurements at two wavelengths was reported by Rogers and Brown (1997). They measured a smoke plume from a large industrial fire with two vertically pointing radars, a UHF profiler (wavelength 33 cm) and an X-band radar (3.2 cm). The smoke was caused by a burning paint factory, located 10 km away from the radars. The factory was burnt to the ground. The plume was overhead at the radars about 20 min after the fire started.

With these wavelengths, purely coherent scattering would give a 37-dBZ difference in the reflectivity factors. A scatterplot of the reflectivity factors shows that the majority of the points lie between the 0-dBZ difference line and the 37-dBZ difference line (see Fig. 4). There is a correlation between the reflectivity factors of both radars. The points lie roughly on a line of slope one-half (a least squares linear fit gives a slope of 0.4). These measurements may also be explained in terms of coherent particle scattering (section 4b). Curiously, a Vaisala CT-12K ceilometer detected little or nothing of the smoke plume.

b. Coherent particle scattering in smoke

Rogers and Brown (1997) investigate two possible explanations for the difference in the observed reflectivity factors: 1) the presence of cm-sized particles and 2) a strongly perturbed structure of atmospheric refractivity.

Large particles: There are indications that the smoke plume contained a small number of large particles. These indications are: a grainy pattern of high-resolution X-band reflectivity, a predominance of downward velocities on the order of 1 m s⁻¹, and the fact that a Vaisala CT-12K ceilometer did not detect the plume. If the particles are so large that the Rayleigh approximation is not valid at X band, then the UHF reflectivity will exceed the X-band reflectivity. Rogers and Brown show that a very small number of centimeter-sized particles can explain the magnitude of the differences that occur if the particles are too large to satisfy the Rayleigh approximation at UHF, but in order to explain the correlation in the measurements the concentration of the particles must increase with their size, which seems unlikely.

It is puzzling that the ceilometer did not detect the plume, because even if the plume contained a small number of large particles, it would probably still contain small particles in much higher concentrations.

Strong clear-air scatter: The heat generated by the fire and gases that are produced could increase the value of C²_n to values considerably higher than that ordinarily observed in the atmosphere, but again it is difficult to explain the correlation.

We suggest that the correlation and the slope near one-half can be explained by coherent particle scattering. For wavelengths of 3.2 and 33 cm, respectively, Eq. (17) becomes

View Expanded

Because the first term on the right-hand side of these equations is a linear function of number density and the second term a quadratic function of number density, there will be some range of values of the number density, where the 3-cm signal is mainly due to incoherent scattering and the 33-cm signal is dominated by coherent scattering. If the number density varies within this range in the plume, then a slope of one-half can be produced on a log–log scatterplot of UHF versus X-band reflectivity. Variations in the other variables could be responsible for the considerable scatter in the measurements shown in Fig. 4. There is some question about the origin of the variations in concentration. They could be due to spatial and temporal variations in turbulent mixing (in which case one may expect considerable variations in the parameters β and L₀). It could also be that the main variations in concentration are due to variations in the amount of smoke generated by the fire.

The order of magnitude of particle density and diameter can be estimated. If β is taken to be unity and L₀ equal to 10 m, then it can be calculated with Eq. (21) that concentrations ranging from 2 to 60 cm⁻³ and a diameter of about 90 μm are needed to explain Fig. 3. This must be considered a very rough estimate, however, because of the uncertainty in the values of the parameters, but the numbers seem reasonable.

It is curious that the ceilometer detected little or nothing of the smoke plume. One would expect this lidar to detect a cloud consisting of particles with the calculated diameters and number densities. It could be that the lidar backscatter from the smoke plume was reduced because the particles were strongly absorbing at the lidar wavelength. Furthermore, the boundaries of the smoke plume are probably much less sharp than those of the bottom of a water cloud, so a significant part of the power may already have been attenuated before the light reached the position with a relatively high backscattering.