1. Introduction
Many numerical weather prediction models, climate models, and limited-area models parameterize cloud processes using bulk water schemes. These schemes carry explicit variables such as cloud water mixing ratio, rainwater mixing ratio, and a variety of mixing ratios defining the classes of ice particles present (e.g., pristine ice, graupel, etc.). However, when a bulk water parameterization is used, the cloud microphysical properties such as the number of ice particles and the extinction coefficient must be diagnosed. This requires an assumption about the microphysical structure of the particles for the cloud and precipitation variables.
Cloud models often use bin microphysical schemes to calculate the particle distributions explicitly (e.g., Reisin et al. 1996) thereby avoiding the need for bulk water parameterizations. However, where bulk water microphysical schemes are used, the cloud models usually carry several classes of ice particles with the small pristine ice crystals developing from either ice nuclei or ice multiplication processes (e.g., Walko et al. 1995). More recently, bulk water schemes have used double-moment schemes to parameterize the ice crystal distributions (e.g., Harrington et al. 1995).
Climate models and weather prediction models require the simplest possible parameterization that captures the critical cloud physical processes. Two important model cloud processes are the interaction of the radiation with the clouds and the role of the clouds in the water cycle. In the model radiation module, the critical microphysical parameter is the extinction coefficient. The water cycle in the model is simplified by dividing the bulk water parameterizations into nonprecipitating and precipitating particles (see, e.g., Rotstayn 1997). In parameterizing both the radiation processes and the water cycle there are implicit assumptions about the particle size distributions for both precipitating and nonprecipitating ice particles. The motivation for the present study was to develop a physically based ice scheme using two bulk ice classes (precipitating and nonprecipitating particles) that treated the precipitation and radiation consistently.
The aim of this paper is to test a new microphysical parameterization that diagnoses the ice particle numbers, the extinction coefficient, and the particle effective diameter from the cloud temperatures and ice cloud mixing ratios in midlatitude cloud systems. A theoretical basis of the parameterization is developed in the second section of the paper. In section 3, the parameterization is tested by using observations of ice particle numbers in frontal clouds observed over southeastern Australia and in cirrus observed during the First International Satellite Cloud Climatology Project (ISCCP) Research Experiment Phase II (FIRE II). Also in this section the calculated extinction coefficients are compared with those from Platt (1997). In section 4, the parameterization is used in a simulation of the passage of a cold front that passed over southern Australia during 17–19 November 1984. The paper concludes with a discussion of the implementation of the new parameterization into a general circulation model (GCM) or limited-area model.
2. Development of a new parameterization
a. Observations of ice particle distributions
The new parameterization is based on a paper by Platt (1997) showing that ice particle concentrations measured in frontal clouds and in cirrus clouds form bimodal distributions (Fig. 1). This bimodal structure of the ice crystal distribution in cirrus clouds has been observed in a large number of studies (e.g., Mitchell et al. 1996). There is some uncertainty in all of these measurements. Problems occur when the forward scattering spectrometer probe instrument is operated in the presence of large drops, whereas there is a lower limit of about 50 μm in the 2D particle measuring probes. Unfortunately, the uncertainties in both instruments coincide with the onset of the second mode in the distribution. The new generation of particle measuring instrumentation should eliminate this problem.
The dataset used by Platt was collected by Heymsfield (1977) using particle measuring system probes. The data were collected in each size channel and binned over a 25-km pass to calculate “pass-average” concentrations. Platt examined the particle size distribution for particles ranging from 2 to 3000 μm. He divided the particle size distribution into three regimes, namely, the small particle regime (2–20 μm), the Heymsfield–Platt power-law regime, and the Marshall–Palmer regime. The mass of particles in the small particle regime was relatively small, while the contribution of the small particles to the extinction coefficient was 30% at −50°C, 8% at −45°C, and 8% at −30°C.
The bimodal structure of the “averaged” ice particle spectra is not fully understood. Mitchell et al. (1996) has suggested that the bimodel distribution is a result of a balance between the nucleation of ice particles and the removal of ice particles by aggregation and enhanced diffusional growth via ventilation. The onset of aggregation and enhanced diffusional growth coincides with the development of the second mode in the distribution.
There are two different approaches to parameterizing this bimodal structure of the ice particle size distribution. Some researchers, such as Mitchell (1994), use two gamma distributions to characterize the two modes of the ice particle distribution. Platt (1997) presents an alternative model characterized by a power law and the simplest gamma distribution often termed the Marshall–Palmer distribution.
Platt (1997) showed that the slope parameter, b, in the power law is a function of temperature (Fig. 3). He estimates that the scatter in b is less that 10% (C. M. R. Platt 1999, personal communication). There are different functional forms of the H–P power law for dry cirrus and for synoptic depressions that give rise to deep winter and springtime ice clouds. Figure 3 shows data from Platt (1997) giving b as a function of temperature for both these cloud types. European Cloud and Radiation Experiment (EUCREX) data from measurements made by the U.K. Meteorological Research Flight (S. Moss, 1998 personal communication) has been added to the figure. It is clear that there is a bifurcation in the data at about −18°C. The dashed line is a fit to the dry cirrus of Heymsfield and Platt (1984) and the EUCREX data, while the solid line fits to the frontal cirrus data of Platt (1997). The data has been interpreted to suggest that below −40°C the frontal clouds revert to dry cirrus clouds. Also included in the figure is a summary of the ice habits at water saturation and ice saturation adapted from Mason (1962). The algorithms for the fitted data are shown in Table 1.
The variation in b as a function of temperature can be interpreted in terms of the changing crystal habits. In the temperature range −3° to −8°C, the principal habits are needles or columns and the observations from Moss suggest that b is about −2.1. There is a step change in b with the onset of the plate regime at about −8°C and the bifurcation takes place at about −18°C. One possible interpretation of the bifurcation is that the dry cirrus clouds from Heymsfield and Platt (1984) and the EUCREX clouds from Moss are subsaturated with respect to water, while the deep frontal ice clouds from Platt (1997) are close to water saturation. This assumption is plausible because of the higher vertical velocities expected in the frontal clouds. Theoretical support for this interpretation of Fig. 3 comes from Rauber and Tokay (1991) and Heymsfield and Miloshevich (1995). Rauber and Tokay (1991) have shown theoretically that frontal clouds are able to sustain a supply of condensate that exceeds ice crystal mass growth rate, thereby allowing the air to reach water saturation. Furthermore, Heymsfield and Miloshevich (1995) have shown that homogeneous nucleation occurs in wave clouds at water saturation once the temperature is below −39°C.
b. Assumptions in the new parameterization
The new parameterization is an extension of Platt (1997) and allows the radiation and microphysics to be treated consistently in a model carrying a bulk water parameterization scheme. The scheme diagnoses the number of ice crystals, the effective diameter, and the extinction coefficient as a function of cloud temperature and cloud mixing ratio. The parameterization is based on the following simple assumptions.
Nonprecipitating-sized particles are assumed to be H–P particles while precipitation-sized particles are assumed to be M–P particles. The contribution of the small particle regime to the development of precipitation and to the extinction coefficient is neglected. However, it should be noted that at very low temperatures this may not be a good assumption.
The b parameter in the power law is different for dry cirrus and frontal cirrus or middle-level clouds.
M–P particles form through an ice phase autoconversion process and increase in mass by a collection process and by direct vapor deposition or decrease in mass by evaporation.
In this paper, cloud systems are defined as either being frontal or dry cirrus in form and vapor deposition on the M–P particles is neglected. This simplification means that the growth of precipitation particles by aggregation is overestimated and vapor growth by H–P particles is also overestimated.
c. Theoretical basis of the ice crystal parameterization
1) Parameterization of ice crystal aggregation
The rate constant used in the model follows Lin et al. (1983) and is defined as R(T) = 0.005 exp[0.025 · (T − 273.16)] where T is the cloud temperature (K). The effect on the model of changing R(T) is to change the rate at which the cloud ice mixing ratio converges with the critical mixing ratio. The critical mixing ratio qcrit is assumed to represent the threshold for aggregation to occur.
The new aspect of the parameterization is that the H–P distribution is used to define wcrit, where wcrit is calculated by integrating the power law between the limits Do and Dc. Here, Do is the size of the smallest ice particle used in the calculation of the ice content and Dc is the upper limit for the power law. Equation (8) then relaxes the cloud ice content back to qcrit.
The fundamental assumption is that the observed average mixing ratios calculated from the averaged H–P distributions can be used to define a lower bound for the mixing ratio characterizing the onset of aggregation for a H–P distribution.
The parameterization depends on Do and Dc and in the standard formulation used in the paper, Do is assumed to be 50 μm and Dc is assumed to be 200 μm. There is very little mass in the range 20–50 μm leading to a change of a few percent in qcrit. A lower limit is chosen because 50 μm is considered to be the lower limit for a reliable comparison of ice crystal numbers with the Australian observations. The sensitivity of the calculation to Do and Dc is discussed later in the paper.
Figure 4 shows the critical cloud ice content, wcrit, calculated as a function of temperature using the observed power laws from Platt (1997). The values of A and b are given in Tables 3 and 4 of Platt (1997). In calculating wcrit, it is assumed that the H–P ice particles are spherical and have a density of 500 kg m−3. Note that wcrit is independent of the air density. The value of wcrit is diagnosed from the fitted lines shown on Fig. 4. Different lines have been fitted to the data for the frontal clouds and the dry cirrus clouds, reflecting the difference in the parameter b in the power law. The algorithms used to calculate the critical ice content are given in Table 2.
The error in the assumed value of b is less than 10%. The error bars show that when an error of 10% in b is used, then the uncertainty in wcrit is about a factor of 5. The solid lines in Fig. 4 were calculated using a regression technique and lie within the error bars of most of the points. The greatest uncertainty is for the points where the temperature is near 42.5°C.
The critical cloud ice content, wcrit, is different if the H–P ice particles are assumed to be crystals and not spheres. The habit ranges as a function of temperature are given in Table 3. However, the number of H–P crystals is independent of the ice crystal habit. Equation (9) is used to transform the critical ice water content of air with spherical ice particles to the critical ice water content of the air with ice crystals.
2) Number of H–P particles
In applying this parameterization in a model calculation, qc is a variable carried by the model. However, in the section of this paper validating the parameterization against Australian observations, the values of both A and b from Platt (1997) are used.
3) Number of M–P particles
In the current set of experiments, Dp is assumed to be independent of temperature and given a value of 400 μm, that is, in the middle of the range suggested by Platt. Here, Dp is probably a function of temperature, although the observations of Platt (1997) are not sufficient to parameterize Dp. The spherical M–P particles are assumed to have a mean bulk density of 100 kg m−3 following Lin et al. (1983) and Rutledge and Hobbs (1983), while the crystal M–P particles are assumed to have a bulk density of 800 kg m−3.
4) Extinction coefficient and effective diameter
3. Comparison with observations
a. Ice crystal numbers
With the onset of the generation of precipitation sized particles, the autoconversion parameterization relaxes the mixing ratio of ice crystals toward qcrit. Near cloud top the number of precipitation-sized particles should be relatively small, and, therefore, the average number of ice crystals at cloud top should approximate the number of H–P particles calculated using Eq. (9). At lower levels in the cloud this assumption may be invalid because precipitation particles are removing cloud particles through the processes of aggregation and accretion.
Equation (9) and the values of A and b used by Platt (1997) have been used to calculate the temperature dependence of the number of H–P particles in frontal and cirrus clouds. The calculated number of H–P particles in frontal clouds and in dry cirrus clouds in the temperature range −7.5° to −57.5°C are shown by the solid and dashed lines, respectively, in Fig. 5.
Figure 5 also shows observed numbers of ice crystals at cloud top in frontal and cirrus clouds for comparison. The observations for frontal clouds are from Mossop (1968), Ryan et al. (1985), and J. Jensen (1994, personal communication). All these clouds were associated with frontal systems crossing southeastern Australia. The cirrus observations shown in Fig. 5 are from Leaitch et al. (1992) and Heymsfield and Miloshevich (1995). The ice crystal numbers reported by Leaitch et al. (1992) are the mean of observations reported by several authors using data from Japan, the North Sea, eastern and western Europe, and as such they may not necessarily be observations at cloud top. The cirrus data near cloud top are from Heymsfield and Miloshevich and were measured during FIRE II.
The number of H–P particles diagnosed for both the frontal clouds and the dry cirrus clouds fits the observed data reasonably well. Between −7.5° and −17.5°C the numbers of H–P particles diagnosed using the frontal parameterization are very similar to the number of ice crystals found in the Australian frontal cloud systems. From −22.5° to −27.5°C the observed ice crystal numbers exceed the diagnosed ice crystal numbers. The number of H–P particles diagnosed using the cirrus parameterization are consistent with the cirrus observations of Heymsfeld and Miloshevich (1995), although the cirrus comparison is over a relatively narrow temperature range. Both the frontal and the cirrus observations represent datasets that are completely independent of each other and independent of the data sets used in formalizing the ice crystal diagnostics.
b. Extinction coefficient and effective diameter
The extinction coefficient and the effective diameter are calculated using Eqs. (12) and (13). These parameterized values have been compared with the calculations from Platt (1997). To ensure consistency in the comparison, the parameters A and b used in the H–P equation were taken from Platt (1997, Table 5), and No and λ in the M–P equation were taken from Platt (1997, Table 5). The limits of integration, Do and Dc for the H–P range and Dp and Dm for the M–P range, also were taken from Platt (1997, Table 2). Calculations were made assuming that (i) all particles were spherical and (ii) all particles were either plates or columns as shown in Table 3.
The equations shown in the appendix were used to calculate Ntc, Ntp, qc, qp, σc, and σp for both spheres and crystals. Figure 6 shows the calculated extinction coefficient (σc + σp) as a function of the calculated ice water content. The point marked “N” is for needles (−7.5°C), the points marked “P” are for plates (−12.5°C, −17.5°C), and the points marked “B” are for bullets (−22.5° to −47.5°C). The assumed H–P and M–P particle density was 800 kg m−3. The points marked “S” are for spherical particles with an assumed H–P density of 800 kg m−3 and an assumed M–P density of 100 kg m−3. These data are superimposed in Fig. 6 from Platt (1997). The solid line is a linear fit to the frontal cloud data as found by Platt (1997) and the dashed line is for tropical cirrus and is taken from Heymsfield and McFarquhar (1996). The other observations are from Kinne et al. (1992), Arnott et al. (1994), and Francis et al. (1994).
The fitted data for tropical cirrus from Heymsfield and McFaquhar (1996) and midlatitude systems (Platt 1997) are very similar. The data for the bullets are close to, but slightly below, those calculated by Platt (1997). The two plate data points lie above Platt’s linear fit and there is a suggestion that the plates have a different slope than that of the bullets. There is also a suggestion of this in Platt (1997, Fig 5). The needle data point lies above the Platt linear fit.
The spherical data points marked S lie above the Platt line, but in the range of the observed extinction coefficients. The reason that the spherical data lie close to the observed data is because the calculated masses are dominated by the M–P distribution and the density of 100 kg m−3 scales the masses by a factor of 8. The calculations show that spherical ice particles with density of 100–200 kg m−3 give a similar extinction coefficient to ice crystals with the same mixing ratio and a density of 800 kg m−3. When a density of 800 kg m−3 is used with the spheres, the ice water content increases, but the calculated extinction coefficient lies below the extrapolated line from Platt (1997). This demonstrates that when ice particles and ice spheres have the same density and same ice water content, the extinction coefficient for the ice crystals is greater than the extinction coefficient for the ice spheres.
Figure 7 shows the diagnosed effective diameters for the ice crystals. Data from Francis et al. (1994) are also included in Fig. 7 showing that the parameterized effective diameters for the crystals are consistent with the observed data.
4. Application of the parameterization to the simulation of the passage of a cold front
A detailed description of this frontal case using a bulk microphysics scheme currently implemented in the Commonwealth Scientific and Industrial Research Organisation (CSIRO) Division of Atmospheric Research Limited Area Model (DARLAM) is given in Katzfey and Ryan (1997).
The model is a two-time-step, semi-implicit hydrostatic primitive equation model with semi-Lagrangian advection on a C grid. The model was run at 30-km grid spacing in the horizontal with 18 equally space levels in the vertical. The time step in the calculation was 120 s. The model has a full physics package with a cloud scheme that has both cloud particles (water or ice) and precipitation (rain or aggregates) with the phase of the particles dependent on the in-cloud temperature. The scheme has standard parameterizations for autoconversion, collection, and evaporation rates (see Cotton and Anthes 1989). The novel aspects of the autoconversion formulation and the diagnostics for H–P and M–P particle numbers, and extinction coefficient have been discussed in previous sections of the paper. Note, that in its present formulation mixed phase clouds are not permitted; to form, that is, all cloud water at grid points above the freezing level is completely frozen once cloud top is colder that −15°C. It is assumed that the mixed-phased clouds are all subgrid-scale convective clouds. The justification of this simplifying assumption is that the observations showed that the supercooled regions were very short lived (King 1982). Any cloud water that was generated was immediately converted to ice by the riming process as evidenced by some of the precipitation-sized particles. Rotstayn et al. (2000) have developed a new parameterization that treats the mixed-phase processes in clouds and tested it using this case study. The simulation showed that the liquid water was unable to be sustained by the cloud system in this case study.
Katzfey and Ryan (1997) showed that the model simulation captures the main features of the front that passed over southeastern Australia from 17 to 19 November 1984. The NOAA-7 satellite and radar imagery shows a complex frontal transition zone ahead of the main front (Fig. 8a). The cloud cover generated by the model has the same finger like structure over the continent with a large prefontal cloud mass to the south (Fig. 8b).
In the present case study, it is assumed that the M–P distribution is made up of spherical particles. King (1982) found that in the deep Australian frontal clouds with tops cooler than −20°C, the larger ice particles seen by the 2D Knollenberg optical probe were “fairly nondescript in appearance with sufficient riming to give a circular appearance.” Whereas when cloud tops were warmer than −20°C, more classical shapes were found.
The standard simulation assumes that all the H–P particles are spherical with the lower limit of the H–P distribution assumed to be 50 μm. A second simulation has been repeated with the H–P particles with a parameterized ice crystal habit. A third series of sensitivity simulation has been carried out in order to investigate the sensitivity of the simulation to the lower limit of the H–P distribution with Do set to 20 and 2 μm. In order to explore the sensitivity of the model diagnostics to the range of ice mixing ratios generated by the model, the output of a single grid point was saved at each time step. The grid point is located close to −39°S, 135°E.
a. Spherical H–P ice particles
The model values of qc and qcrit at −39°S, 135°E are shown as a function of temperature in Fig. 9a. The model values of the mixing ratio lie above the critical mixing ratio line. Where qc is less than qcrit, there is no in situ production of precipitation particles by autoconversion. However, while there are no precipitation particles generated by the autoconversion process, the ice content of the cloud may still be reduced by the collection of H–P particles by M–P particles falling into the grid box. Where qc exceeds qcrit, there are losses of cloud ice in the grid box from the combined effects of the autoconversion and collection processes. The simulation is dependent on the choice of the rate constant used for the autoconversion. Figure 9b shows the effect of increasing the rate constant by an order of magnitude. As expected, the model mixing ratios now lie much closer to qcrit.
The total number of particles (H–P + M–P) as a function of temperature is shown in Fig. 10a. A comparison of H–P and M–P particle numbers shows that in the top two model levels containing cloud, the total number of particles is dominated by the H–P particles. However, at the lower model levels the M–P particles dominate the particle size distributions. The maximum concentration of particles in a cloud is of the order of 3 × 105 m−3 or 300 L−1 at about − 17°C. Note that in Fig. 10a all particles warmer than 0°C are water, while particles colder than 0°C are ice. The effect of increasing the autoconversion rate by an order of magnitude reduces the number of ice particles in the top two model levels from 80 L−1 to about 6 L−1 (Fig. 10b). However at the lower levels, where the number of precipitation particles becomes important, there is only a small change in the total number of ice particles.
Figure 11 shows the model-calculated extinction coefficient, using Eq. (12), as a function of ice or liquid mixing ratio. There are three sets of model output points in Fig. 11 labeled cloud water, cloud ice, and rain. The cloud water points are generated by the low-level stratiform cloud that forms at the end of the simulation and lie close to the theoretically calculated values for 15-μm diameter droplets. The rainwater points are dominated by the M–P particles. The cloud ice points have the most variance and for the same ice water content the extinction coefficient is less than the cloud water data and greater than the rainwater data. The density difference between the water (1000 kg m−3) and the M–P particles (100 kg m−3) can explain most of the difference between the rain points and the cloud ice points.
Figure 12a shows the optical depth diagnosed from ISCCP and Fig. 12b shows the modeled optical depths at 0600 UTC. The agreement between the satellite imagery and the model is encouraging for the regions where there is deep frontal cloud. Near 138°E the microphysical parameterization generally underestimates the optical depths over the land and over the ocean between 34° and 40°S while farther to the south the model overestimates the peaks over the ocean. One explanation for the deficiency is that over land the observations and the model both show there is significant convective activity. In the present study, the effect of the subgrid-scale cloud on the optical depth has not been considered. It is also known that the model poorly simulates the thin layers of midlevel and cirrus cloud that occur well ahead of the front. The stronger observed optical depth peaks coincide with regions of heavy rain indicating that the rainfall being produced in these regions is probably excessive.
b. Sensitivity to ice crystal habits and integration limits
The model simulations have been repeated using the parameterized ice crystal habits for the H–P particles, but the M–P particles were still assumed to be spherical particles (Fig. 13a). The simulation showed an insignificant difference in the calculated extinction coefficients south of 40°S, but an increase in extinction coefficient in the lighter rainfall region north of 40°S. The ice crystal approximation is in better agreement with the ISCCP data than the spherical ice crystal approximation. These results tend to confirm the interpretation that in regions of heavy precipitation the extinction coefficient is dominated by the M–P distribution. This is consistent with the simple calculations based on the data from Platt (1997) and is also consistent with the conclusions of Heymsfield and McFarquhar (1996).
The parameterization of the H–P particle numbers and the H–P component of the extinction coefficient is dependent on the limits of the integration Do and Dc. In the base simulation Do was set to 50 μm to give ice crystal numbers that were consistent with the observations of Ryan et al. (1985). However, as pointed out by Platt (1997), the lower limit of the power law extends to 20-μm diameter while ice crystal as small as 2-μm diameter are observed.
The simulations have been repeated for spherical ice crystals setting Do to 20 and 2 μm, respectively. The difference maps for the optical depths in these simulations are shown in Figs. 13b and 13c. The changes between the standard run with the 50-μm cutoff and the run with a 20-μm cutoff gives a small increase in the optical depth of the order of 4% (the increase in optical depth varies by about 1 in 25). The comparison between the simulation with the 50-μm cutoff and the simulation with the 2-μm cutoff is much larger and is comparable to the difference in optical depth between the ice crystals and spheres.
The 50-μm simulation has been repeated using the H–P ice crystal distribution and the parameterization for Dc. The differences in the optical depths between this simulation and the standard simulation are shown in Fig. 13d. There is very little difference in Fig. 13d indicating the parameterization of Dc has a relatively minor impact on the optical depth.
The sensitivity of the calculation to the limits of the Marshall–Palmer distribution has been discussed earlier and Eq. (11) has been used to parameterize Dm as a function of temperature. It is also likely that the upper limit in the Marshall–Palmer calculation is also a function of temperature and the total ice content. However, the author was unable to obtain sufficient data to formulate such a parameterization. The 400-μm value used in the standard calculation is in the middle of the range found by Platt (1997).
Finally, the simulation has been repeated using the ice crystal parameterization for the Marshall–Palmer distribution with ice particle densities of 800 kg m−3 As shown in Fig. 6, the extinction coefficient for crystals with a density of 800 kg m−3 is of the same order as spheres with a density of 100 kg m−3. Consequently the model simulation using the crystals made only a minor difference to the diagnosed optical depths.
5. Discussion and conclusions
The case study presented here shows the following.
The new parameterization allows ice crystal numbers and optical properties to be diagnosed in models that use a bulk microphysical parameterization. The ice particle numbers and the optical depth are two important properties that can be validated against observations. The new parameterization unifies the radiation and precipitation branches of the microphysical package.
The ice crystal numbers and optical properties diagnosed are in reasonable agreement with the observations in this case study.
The parameterization of the H–P and M–P particles has been extended to include nonspherical ice crystals.
The parameterization is sensitive to the lower limit for the long crystal dimension, Do, for the H–P distribution. For the optical properties the lower limit Do is probably best set at 20 μm based on the observations of Platt (1997). However, for calculation of ice crystal number the lower limit of 50 μm probably gives results that are more consistent with the Australian observations.
The parameterization is not overly sensitive to the upper limit for the long dimension, Dc, of the H–P distribution, although the data from Platt (1997) can be used to set Dc as a function of temperature.
The sensitivity of the parameterization to the lower limit of the Marshall–Palmer distribution, Dp, is also probably a function of temperature but this sensitivity has not been tested because of the unavailability of data to formulate a realistic parameterization.
The number concentrations are dominated by the H–P component of the total size distribution (H–P + M–P), and the climatological observations of King (1982) and Ryan et al. (1985) suggest that the simulated ice crystal numbers in the prefrontal cloud are realistic. The variables in the diagnostic for the ice particle numbers that require observational verification are the assumed ice particle densities for the H–P and M–P distributions, limits of integration, and the parameters b and λ.
The diagnostic for the extinction coefficient allows for the crystal habit in the H–P and M–P ranges. In the present modeling study, the justification for the assumption of spherical particles in the M–P distribution is based on King (1982) who found that in the deep Australian frontal clouds the larger ice particles were fairly nondescript in appearance with sufficient riming to give a circular appearance. These observations show that there are some cases where the spherical assumption is reasonable while in other cases the ice habit of precipitation particles needs to be included in the M–P parameterization. In the modeling case study there are no observations of the density of the M–P particles. The assumed very low density is based in the assumption that they were aggregates.
The model diagnostics for the extinction coefficient compared well with Platt (1997). Both parameterizations diagnose the extinction coefficient from the ice mixing ratio. Platt (1997) uses a power-law relationship to relate the extinction coefficient to the ice mixing ratio. This parameterization assumes a crystal habit and an ice crystal density that is a function of ice particle size for both the H–P and M–P components of the distribution. The Platt parameterization is currently being used in the CSIRO GCM cloud parameterization scheme (Rotstayn 1997, 1999). The diagnostic developed in this paper calculates the extinction coefficient as a function of the ice mixing ratio, which can be expressed in terms of an effective diameter of the ice particles in the cloud.
The parameterization of the precipitation processes and the optical properties are internally self-consistent in the new formulation. However, the parameterization raises the question of what is the rate constant for the autoconversion coefficient in the chain of microphysical ice processes. Unlike the autoconversion process for cloud droplets, there is no physical basis for setting the rate constant for the ice process. This is a fundamental cloud physics problem and is beyond the scope of this paper.
There is a need for more careful observations to provide a more physical basis for the choice of constants used in the parameterization. A careful analysis of more microphysical data from different cloud systems is required so that the assumption of the functional form of the power-law parameter, b, can be generalized. More observations are required to validate the functional form of wcrit, and there is a need for observations in the M–P range to give a more physical base to the choice of Dp.
The new parameterization has been tested in only one case study. The case study is of a cloud system where there were very few mixed phase clouds (King 1982). The new parameterization will be further tested in simulations using the CSIRO limited area model, DARLAM as part of the activities of Global Energy and Water Cycle Experiment Cloud System Study Working Group III. The aim of these modeling studies is to develop new parameterizations for weather and climate models through the use of cloud resolving models (Browning et al. 1993).
The parameterization can be applied in climate models where both dry cirrus and frontal systems are active. An objective criterion for determining whether to call the dry cirrus algorithm or the frontal cloud algorithm would be whether or not the model atmosphere was close to water or ice saturation. The simplest formulation for this trigger is to calculate the vapor deposition at water saturation on the ice particles in the H–P and M–P distributions as a diagnostic. If the vapor deposited by adjustment to water saturation exceeds the diagnostic calculation, then the frontal algorithm is used and the cloud mixing ratio is set to water saturation. If the vapor is less than the diagnostic calculation, then the dry cirrus algorithm is used. In frontal clouds riming takes place and the M–P particles are assumed to be spherical, while if there is no riming the M–P particles are assumed to be crystals. This formulation is to be tested in the CSIRO GCM.
Acknowledgments
This paper was stimulated by discussions with Martin Platt and Leon Rotstayn. I am grateful to Sarah Moss from the UKMO for sending the unpublished data used in Fig. 3. I would also like expresses my thanks to Martin and Leon for their suggested changes to the paper and to Joanne Richmond for her assistance in drafting the figures. Finally, I would like to thank the reviewers for three constructive and thought provoking reviews. The paper is a contribution to the CSIRO Climate Change Research Program.
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APPENDIX
Derivation of H–P and M–P Distributions
H–P distribution: Particle numbers, mixing ratio, and extinction coefficient
Spheres
Plates
Columns
Generalized form
The H–P mixing ratio and the extinction coefficient are generalized as follows. In the generalized expressions, D refers to the maximum particle dimensions. Table 3 shows the shape parameters for spheres, plates, and columns as function of temperature based on data from Heymsfield (1972).
M–P distribution: Particle numbers, mixing ratio, and extinction coefficient
Spheres
Here Γ(a) is the gamma function and P(a, x) is the incomplete gamma function (Abramowitz and Stegun 1972).
Generalized form
The M–P mixing ratio and extinction coefficient can be generalized using the ice crystal shape factors discussed above. In the generalized expressions, D refers to the maximum particle dimensions.
Observations of cloud particle number density N(D) as a function of the long dimension for clouds sampled in the temperature interval −25° to −30°C (from Platt 1997).
Citation: Journal of the Atmospheric Sciences 57, 9; 10.1175/1520-0469(2000)057<1436:ABPOTI>2.0.CO;2
The observed exponent λ (m−1) for the Marshall–Palmer distribution in clouds as a function of temperature. Here “P” (Passerelli 1978) is from IL, “HHHP” (Houze et al. 1979) is from WA;SMPC (Stewart et al. 1983) and “GM” (Gordon and Marwitz 1986) are from CA, “PLATT” (Platt 1997) is from the United States; AWSE (J. Jensen 1994, personal communication) is from Australia, “YL” (You and Liu 1989) is from China, and “B” (Bennetts and Ryder 1984) and “M” (Marecal et al. 1993) are from Europe. The bold line labeled “RYAN” is the parameterization for λ used in this paper.
Citation: Journal of the Atmospheric Sciences 57, 9; 10.1175/1520-0469(2000)057<1436:ABPOTI>2.0.CO;2
The slope parameter b as a function of temperature. The data points are from Platt (1997), Heymsfield and Platt (1984), and Moss (1994, personal communication). Here, b is parameterized by fitting the dashed line for cirrus clouds and the solid line for frontal clouds at temperatures between −18° and −37.5°C to the observations. The crystal habits at ice and water saturation are shown at the top of the figure.
Citation: Journal of the Atmospheric Sciences 57, 9; 10.1175/1520-0469(2000)057<1436:ABPOTI>2.0.CO;2
Critical ice content wcrit (kg m−3) as a function of temperature. The solid circles are from Platt (1997) and the open circles are from Heymsfield and Platt (1984) assuming the ice particles to be spheres. Here wcrit is parameterized by fitting the solid line for spheres and the dashed line for ice crystals. The upper line in for both the spheres and the ice crystals is the regression line for the H–P data and the lower line is for the P data.
Citation: Journal of the Atmospheric Sciences 57, 9; 10.1175/1520-0469(2000)057<1436:ABPOTI>2.0.CO;2
Ice crystal numbers (no per liter) as a function of cloud-top temperature. The observations for frontal clouds are “M” (Mossop 1968), “K” (Ryan et al. 1985), and “J” (J. Jensen 1994, personal communication). All these clouds are for frontal systems crossing southeastern Australia. The cirrus observations are from “L” (Leaitch et al. 1992) and “H” (Heymsfield and Miloshevich 1995). The diagnosed ice crystals numbers in the frontal cloud (solid line) and cirrus clouds (dashed line) are calculated using qcrit in Eq. (8).
Citation: Journal of the Atmospheric Sciences 57, 9; 10.1175/1520-0469(2000)057<1436:ABPOTI>2.0.CO;2
Volume extinction coefficient as a function of ice content. The points marked N are needles (−7.5°C), the points marked P are plates (−12.5°C, −17.5°C), and the points marked B are bullets (−22.5° to −47.5°C); the points marked “S” are for spherical particles. The data from Kinne et al. (1992), Arnott et al. (1994), and Francis et al. (1994) are taken from Platt (1997, Fig. 6). The solid line is the extinction coefficient for frontal ice clouds calculated Platt (1997) and the dashed line is from Heymsfield and McFaquhar (1996).
Citation: Journal of the Atmospheric Sciences 57, 9; 10.1175/1520-0469(2000)057<1436:ABPOTI>2.0.CO;2
Effective diameter as a function of cloud temperature. The diagnosed effective diameters are calculated using Eq. (13). The closed circles are calculated for ice crystals and the closed triangles are data from Francis et al. (1994).
Citation: Journal of the Atmospheric Sciences 57, 9; 10.1175/1520-0469(2000)057<1436:ABPOTI>2.0.CO;2
(a) Cloud cover and (b) model cloud cover at 0600 UTC 18 Nov 1984. The contours represent fractional cloud cover.
Citation: Journal of the Atmospheric Sciences 57, 9; 10.1175/1520-0469(2000)057<1436:ABPOTI>2.0.CO;2
(a) Ice–cloud water mixing ratio qc sampled every 6 h at −41°S and 139°E as a function of cloud temperature at all model levels. The crosses show the model values and the solid line shows the critical liquid water used in the autoconversion parameterization. (b) Ice–cloud water mixing ratio qc when the autoconversion rate for H–P particles has been increased by a factor of 10.
Citation: Journal of the Atmospheric Sciences 57, 9; 10.1175/1520-0469(2000)057<1436:ABPOTI>2.0.CO;2
(a) The total number of cloud particles (Ntc + Ntp) diagnosed from the model run as a function of cloud-top temperature when sampled every 6 h at −41°S, 139°E for the spherical ice crystal habit. (b) The total number of cloud particles (Ntc + Ntp) diagnosed from the model run as a function of cloud-top temperature when the autoconversion rate for H–P particles has been increased by a factor of 10.
Citation: Journal of the Atmospheric Sciences 57, 9; 10.1175/1520-0469(2000)057<1436:ABPOTI>2.0.CO;2
The volume extinction coefficient as a function of total ice–cloud water mixing ratio (qc + qp) when sampled every 6 h at −41°S, 139°E. The solid lines marked spheres and crystals are from Fig. 7. The line marked droplets are the calculated values for a monodispersed distribution of 15-μm diameter. The model extinction coefficient for cloud drops, raindrops, and ice particles are clearly distinguished.
Citation: Journal of the Atmospheric Sciences 57, 9; 10.1175/1520-0469(2000)057<1436:ABPOTI>2.0.CO;2
(a) The ISCCP GMS-3 DX cloud optical depth and (b) the model optical depth at 0600 UTC on 18 Nov 1984. The contours of cloud optical depth are in units of 20 starting with 5.
Citation: Journal of the Atmospheric Sciences 57, 9; 10.1175/1520-0469(2000)057<1436:ABPOTI>2.0.CO;2
(a) The difference between the optical depth for ice crystals and ice spheres (Do at 20 μm and Dc at 200 μm). The contours for cloud optical depth are in units of 5. (b) The difference between the optical depth (Do at 20 μm and Dc at 200 μm) and optical depth (Do at 50 μm and Dc at 200 μm) at 0600 UTC on 18 Nov 1984. The contours for cloud optical depth are in units of 1. (c) The difference between the optical depth (Do at 2 μm and Dc at 200 μm) and optical depth (Do at 50 μm and Dc at 200 μm) at 0600 UTC on 18 Nov 1984. The contours of cloud optical depth are in units of 5. (d) The difference between the optical depth optical depth [Do at 20 μm and Dc from Eq. (14)] and optical depth (Do at 50 μm and Dc at 200 μm) at 0600 UTC on 18 November 1984. The contours of cloud optical deth are in units of 1.
Citation: Journal of the Atmospheric Sciences 57, 9; 10.1175/1520-0469(2000)057<1436:ABPOTI>2.0.CO;2
The algorithm for the slope parameter, b, and the cloud type and temperature range (K) over which the algorithm is valid.
The algorithm for the slope critical ice water content, qcrit (kg kg−1), and the cloud and crystal types and temperature range (K) over which the algorithm is valid.
The crystal shape parameters and the crystal type and temperature range (K) over which the parameter is valid.
The H–P constants, B1 to B10 for spheres, plates, and columns.
The M–P constants, C1 to C10 for spheres, plates, and columns.
