Cirrus Cloud Simulation Using Explicit Microphysics and Radiation ...

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JOURNAL OF THE ATMOSPHERIC SCIENCES

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Cirrus Cloud Simulation Using Explicit Microphysics and Radiation. Part II: Microphysics, Vapor and Ice Mass Budgets, and Optical and Radiative Properties VITALY I. KHVOROSTYANOV Central Aerological Observatory, Dolgoprudny, Moscow Region, Russian Federation

KENNETH SASSEN Department of Meteorology, University of Utah, Salt Lake City, Utah (Manuscript received 25 March 1997, in final form 5 September 1997) ABSTRACT The 2D/3D cloud model complex with explicit microphysics and radiation described in Part I is used to simulate the development of a midlatitude cirrus cloud, including interaction with radiation. To account for the effects of the interaction of various scales of motion on cloud development, a synoptic-scale vertical velocity field is superimposed on the mesoscale velocity field generated by the model, mimicking the effects of an upperlevel shortwave trough. The main results under the conditions simulated here are the following. Cirrus cloud growth is much slower than assumed previously, because the process of vapor deposition to ice crystals is far from instantaneous: the crystal phase relaxation time (i.e., the characteristic time of vapor absorption by crystals) takes 0.5–2.0 h. Even after 1 h of cloud development, supersaturation with respect to ice can remain 5%–10%, while the condensed ice is only 40%–60% of the amount that would be realized assuming that all excess vapor is transformed into ice in typical model time steps. Although experimental and theoretical studies have produced widely divergent longwave mass absorption 2 21 coefficients aabs , model results show that a single ‘‘representative’’ value of m , ranging from 100 to 3500 cm g aabs is inappropriate. Vertical profiles typically exhibit values of ;800–1000 cm 2 g21 in the upper cloud region m containing the smallest particles, in contrast to ;100–300 cm 2 g21 for the larger crystals in the main cloud. The optical scattering coefficients behave similarly, with typical values of ;2000–2500 cm 2 g21 in the upper cloud regions and ;300–500 cm 2 g21 in the lower cloud regions. A strong horizontal variability is also a characteristic feature of these coefficients. 2 21 Many GCM and climate models use seemingly overestimated aabs ). Sensitivity m values (e.g., 1000 cm g tests show that the use of such values increases cooling in the upper cloud and heating in the lower cloud, which can lead to an unwarranted increase in upper-tropospheric static instability. The postulated effects of the positive feedbacks between clouds and greenhouse gas–induced global warming would likely be different in magnitude (or in sign) if the more realistic approach of using cloud microstructure–dependent absorption and scattering coefficients could be adopted. Consideration of microphysics also shows that the decrease in the shortwave radiative balance (albedo effect) in the simulated midlatitude cirrus cloud exceeds the net gain in the longwave balance (greenhouse effect) near midday, due to the abundance of relatively small crystals in the upper cloud region where cloud regeneration is taking place.

1. Introduction In the first part of this series (Khvorostyanov and Sassen 1998), hereafter refered to as Part I, we have presented a description of a 2D/3D mesoscale cloud model using explicit microphysics and radiation, with particular emphasis on the numerical techniques required for the proper simulation of midlatitude cirrus clouds. Improving our knowledge of the formation and

Corresponding author address: Kenneth Sassen, 819 Browning Bldg., University of Utah, Salt Lake City, UT 84112. E-mail: [email protected]

q 1998 American Meteorological Society

growth of cirrus cloud systems is important to help settle a variety of climate and climate-change related issues, particularly through allowing the introduction of more reality into large-scale model research. The problem of natural and human-induced climate variability has traditionally been studied by large-scale models (GCMs), while the mesoscale cloud modelers have paid much less attention to the climate problem. So, a sort of a gap has existed for about 30 years between GCMs and cloud models, both in purpose and in the methods of cloud modeling or parameterization. It can be said that these two types of models (i.e., large scale and mesoscale) have been developing in parallel, but

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KHVOROSTYANOV AND SASSEN

almost independently. However, recognizing the need for more concerted cloud modeling activities to address climate issues, at the beginning of the 1990s as part of the Global Energy and Water Cycle Experiment (GEWEX), a special program was organized under the auspices of the World Climate Research Program (WCRP). The GEWEX Cloud System Study (GCSS) is intended to join and coordinate the efforts of mesoscale cloud modelers and observationalists to improve our understanding of cloud properties and cloud–radiation interactions, in order to develop better cloud parameterizations for application to GCMs and weather forecast models (Browning 1994). Within the four GCSS Working Groups, WG-2 is devoted to cirrus and midlevel clouds.1 First, both types of cloud models can be tested and verified against experimental data, and second, the findings from these models can be intercompared for mutual improvement and serving the common purpose—the elaboration of recommendations for GCM parameterizations. The problems associated with the numerical simulation of cirrus clouds are numerous, but in order to properly reproduce the development and life cycle of cirrus it is of paramount importance to account for the dynamic forcing at various scales of motion and comprehend their interactions. It can be appreciated from experimental studies (e.g., Sassen and Dodd 1988; Sassen et al. 1989; Sassen et al. 1990; Starr and Wylie 1990; Heymsfield and Miloshevich 1995) that cirrus clouds are closely related to patterns of the vertical velocity field formed by the superposition of synopticscale events and mesoscale motions of different origin and scale. For example, one of the most well-documented cirrus cloud field studies was observed during the First ISCCP (International Satellite Cloud Climatology Project) Regional Experiment (FIRE) Intensive Field Observations (IFO-I) field campaign on 27–28 October 1986. Starr and Wylie (1990) demonstrated that these cirrus clouds were associated with a large-scale upper-level pattern consisting of a trough riding along a ridge crest. Such troughs are the domain of persistent updrafts (Holton 1979), and in this case their maximum velocities were estimated to be 5–10 cm s21 using rawinsonde data and the adiabatic method (Starr and Wylie 1990), and up to 2–4 cm s21 using the European Centre for Medium-Range Weather Forecasts (ECMWF) and mesoscale RAMS models (Heckman and Cotton 1993). Hence, large-scale (synoptic) dynamical forcing with vertical velocities on the order of a few centimeters per second are often identified as being the main reason for the formation and maintenance of extended cirrus cloud fields. On the other hand, Sassen et al. (1989) and Smith et al. (1990) have shown with use

1 See the GCSS WG2 World Wide Web site at http:// eos913c.gsfc.nasa.gov/gcsspwg2/.

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of spectral analyses of aircraft data that the power spectra of vertical velocity exhibit well-pronounced mesoscale maxima with wavelengths on the order of 1–10 km, thus providing evidence for the mesoscale organization of vertical velocities that could be caused by the static stability of the atmosphere, mesoscale substructures in synoptic patterns, orography, or other reasons. As can be appreciated from the model description in Part I, our model configuration allows us to address this important issue, as the results presented here will illustrate. Below, we apply the model to the simulation of a generic type of midlatitude cirrus cloud occurrence associated with an upper-level trough and weak synoptic-scale ascent. 2. Physical situation and model initialization In this study we simulate an event of cirrus cloud formation and development in the mesoscale but at the same time account for the influence of synoptic-scale vertical velocity Wsyn on cloud formation. The total velocity field in our simulation results from the superposition of Wsyn and the model-generated mesoscale velocities. Because the field of Wsyn is governed by synoptic processes with horizontal scales of 1000–3000 km (Heckman and Cotton 1993) and cannot be produced by this or any other mesoscale model employing horizontal scales of 100–200 km, we have essentially modified the characteristics of the 2D model as compared to the previous versions (Khvorostyanov 1995; Khvorostyanov et al. 1996), as described in Part I. The concept was based on the results of experimental studies revealing characteristic horizontal inhomogeneities in cirrus structure, and their linkage to atmospheric dynamics (Sassen et al. 1989; Sassen et al. 1990; Starr and Wylie 1990; Smith et al. 1990). The computational domain consists of three regions in the vertical: 1) a lower layer of 7 levels (z 5 0–6.0 km) with vertical grid length of 1.0 km; 2) a main (middle) cloud domain of 31 levels (6.0–10.5 km) with 150-m grid length; and 3) an upper layer of 15 levels (10.5–24.5 km) with 1.0km grid length. Such a structure allows the simulation with sufficient vertical resolution of high- and midlevel clouds, as well as the calculation of the longwave and solar radiative fluxes in the layer below (including surface radiative and temperature changes caused by the cloud) and above the cloud (including outgoing longwave and reflected solar radiation). The horizontal extent of this model simulation is 96 km, with 3-km horizontal grid length. The simulated horizontal domain is approximately equal to one horizontal grid cell used in the study by Heckman and Cotton (1993), thus allowing us to investigate the processes that are subgrid and smoothed in models designed for the larger mesoscales. We have superimposed the 2D field (in vertical cross section) of Wsyn in the following way: 1) in the vertical direction, a layer in height from z 5 7.0–9.5 km has a central maximum of 5 cm s21 ,

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FIG. 1. Initial profiles of temperature T (8C), potential temperature Q (K), relative humidity with respect to ice RHI (%), and horizontal wind speed u (cm s21 ).

to simulate an upper-level trough (ULT); 2) in the horizontal, this maximum is located in the region of x 5 30–40 km and decreases to 1.0–2.0 cm s21 at the left boundary and to 0.2–1.0 cm s21 at the right boundary (x 5 96 km), to simulate the finite horizontal extension of the ULT. While mesoscale velocities change in time and space, Wsyn is kept constant during the simulation within the computational domain, which is moving with the mean wind along with the ULT. In other words, the domain is being advected along the mean wind parallel to the ridge crest. In a manner of speaking, we simulate the phenomenon of cirrus cloud ‘‘surfing’’ along the ridge crest, which itself is moving with the synoptic wave. Here the pressure is lowering within the forming upper-level trough, giving rise to the large-scale lifting field and the formation of cirrus.

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In these simulations the initial conditions were based on some typical cases observed during the FIRE IFOI, the 1989 European Cirrus Experiment (EUCREX), and the Zvenigorod cirrus field campaigns that have been carried out since 1986 near Moscow. The inputs were not chosen from any specific case, rather they corresponded to some generalized atmospheric conditions in autumn or spring midlatitudes, as found in the abovementioned cirrus research studies. A similar approach with initial typical model data was used by Starr and Cox (1985a). As shown in Fig. 1, the initial profile of temperature T corresponds to a relatively stable atmosphere with T 5 2238C at z 5 6.0 km, and T 5 2578C at z 5 10.5 km (i.e., at the bottom and top of the main computational domain). The mean lapse rate was G 5 7.558C km21 , although a slightly less stable layer from 8.0 to 9.0 km is clearly seen in the potential temperature Q profile. Initial humidity was supersaturated with respect to ice in the 7.0–9.2-km layer with a maximum supersaturation of 5%. Note that these profiles closely resemble the soundings used by Jensen et al. (1994) for the simulation of a FIRE IFO-I cirrus case study and thus are representative of midlatitude autumn. The horizontal wind speed had a maximum of 4 m s21 in the same layer (i.e., the residual wind speed is given in Fig. 1 after subtracting the mean horizontal wind in our domain, thus representing a Lagrangian approach in the horizontal direction). The coincidence of the maximum wind speed and relative humidity is not accidental, since we assume that the layer with increased humidity is caused by the horizontal advection of moisture from the upwind direction. A similar hypothesis was adopted in the 1D cirrus model by Jensen et al. (1994). Finally, for radiative purposes we use a surface temperature of 11.68C and relative humidity of 70%, a 10% surface albedo (i.e., a water surface), a solar zenith angle of 608 (corresponding to ;538N latitude during October), and a simulation start time of local noon. The boundary conditions were chosen similar to Starr and Cox (1985a): free slip conditions for the stream function and vorticity at the upper and lower boundaries, where also the eddy diffusion terms are specified as zero, while precipitation is allowed through the lower boundary; cyclic boundary conditions were chosen in the lateral direction. Small random temperature perturbations of 0.28–0.38C were imposed at the bottom of the domain, which exponentially decrease upward. The mesoscale vertical velocities that are produced by the model due to the initial temperature perturbations are superimposed on the large-scale velocities in a manner similar to Starr and Cox (1985a). Below, we provide views of the formation and development of the simulated cirrus clouds after periods of 30 min and 60 min, which is often considered in GCMs as a characteristic time of transformation of the vapor excess into ice (e.g., Ramanathan et al. 1983).

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FIG. 2. Vertical cross section of dynamical fields at 30 min (left) and 60 min (right) simulation time: streamfunction c (m 2 s21 ), vertical velocity w (cm s21 ), and supersaturation field with respect to ice DEL I (%).

3. The process of cirrus formation a. Dynamical and microphysical properties 1) 30-MIN

SIMULATION

The development of mesoscale updrafts caused by the imposed temperature disturbances starts at the beginning of the model run, as do the processes of ice crystal nucleation and growth by deposition, accompanied by the release of latent heat. The dynamical features of the atmosphere after 30 min of simulation are shown at the left side of Fig. 2. During the first 10–15 min, quasiperiodic cellular structures similar to those simulated by Starr and Cox (1985b) begin developing, but subsequently the interaction of these cells with the large-scale velocity field, and their modulation by latent heat release, causes their gradual merging. As can be seen in the field of the streamfunction c, two vortices developed after 30 min, resulting in the existence of five areas of updrafts with vertical velocities w displaying

maxima of 2.5–6.0 cm s21 (hereafter, vertical velocities include Wsyn ). Since we assumed horizontal homogeneity along the y axis (i.e., perpendicular to our domain), these regions of updrafts represent a vertical cross section of the cirrus bands. Note that these disturbances are caused by the perturbation in the temperature field but are smoothed somewhat due to the rather gross horizontal grid step of 3 km. The main region of updrafts is located in the horizontal domain at x 5 18–45 km, and at heights z 5 7.0–9.0 km where mesoscale updrafts are reinforced by the large-scale ascent. This accelerates the processes of crystal growth and latent heat release, and increases the local buoyancy. Vertical velocities exhibit maxima in the layer from z 5 8.0 to 9.0 km with a correspondingly increased thermal instability (see the potential temperature profile in Fig. 1). As the convective process is limited in the vertical by the more stable stratification aloft and beneath, compensating downdrafts are caused

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FIG. 3. Vertical cross section of microphysical properties after 30 min (left) and 60 min (right) of cirrus development: crystal number density N I (L21 ), IWC (mg m23 ), and crystal mean equivalent radius r I (mm).

on both sides of this area. Vertical velocity decreases or even becomes negative in the region x 5 8–16 km, and a wide region of downdrafts (up to 23 cm s21 ) develops downwind (i.e., to the right) at x 5 46–56 km. Farther downwind, the impact of the large-scale ascent decreases, but three mesoscale updraft cells caused by the initial temperature disturbances are clearly seen in Fig. 2b. Figure 2 clearly shows that the maximum dynamical disturbances along the horizontal direction develop in the central portion of the computational domain and decrease toward the top and bottom. This feature of an upper-tropospheric layer with cirrus was observed during FIRE IFO-I, as analyzed and described by Sassen et al. (1989) and Smith et al. (1990). Thus our results are in qualitative agreement with the above-mentioned studies. After 30 min of cloud development the supersaturation with respect to water is negative everywhere in the domain: that is, the formation of new cloud particles

tends to be suppressed. Supersaturation with respect to ice (d i in Fig. 2) is positive from z 5 6.8 to 9.2 km. It has a maximum of ;10% at around 8.0 km in the layer with maximum vertical velocity, which is the main factor in the production of supersaturation. It is important to note that despite the 30-min cirrus development time, d i is not equal to zero. This is obviously in contradiction with some bulk models where all the excess of the vapor above saturation is transformed almost instantaneously into ice. The relatively large value of predicted supersaturation is a consequence of the explicit microphysical approach used in this model, as is discussed in detail below with the aid of Fig. 3. Figure 3 (left) depicts the meso- and microscale structure of the simulated cirrus in terms of the crystal number density N I (L21 ), ice water content IWC (mg m23 ), and crystal mean radius r I (mm). Crystal number density N I exhibits some slight cellular structure with maximum concentrations of 100–140 L21 at z ; 9.0 km, which,

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as suggested by Sassen and Dodd (1988, 1989), Heymsfield and Sabin (1989), and DeMott et al. (1994), is caused mostly by the ice nucleation process of the homogeneous freezing of haze particles below water saturation. We note that the maximum N I and its location resemble those obtained in the 2D model run reported by DeMott et al. (1994) with a similar hypothesis of ice crystal nucleation. Undoubtedly, variations in the treatment of the ice nucleation mechanism will change the crystal concentrations and size distributions, but since our main purpose here is to follow the further development of the cirrus cloud and its radiative properties, we will put off consideration of these processes for future studies. The maximum IWC at this time reaches 3–6 mg m23 and is located ;1 km below the level of maximum N I . In the horizontal direction, the maximum IWC coincides mostly with the region of the strongest updrafts (x 5 15–45 km) and spreads downwind where it smoothly decreases. No patchy or cellular structure is seen in the IWC field, in contrast to the vertical velocity field: in particular, there is no sharp IWC decrease in the region of downdrafts at x 5 48–56 km. This further indicates that the processes of crystal growth–evaporation do not follow immediately after vertical velocity–supersaturation variations, such that there is some time lag determined by the crystal phase relaxation time. It is appropriate to term the layer around 9.0-km height the crystal generation layer, while the layer beneath at ;8.2 km can be called the IWC generation layer, or deposition layer. Crystal generation is the highest in the upper layer because of the rapid nucleation rate at the coldest temperatures, and crystal growth is the fastest in the layer below where the mean vertical velocities and supersaturations are the largest. In this example, these two layers are separated somewhat, but generally their mutual location should be determined by the location of the regions with the coldest temperatures and maximum vertical velocities. The mean equivalent crystal radius r I (Fig. 3, bottom), displays minimum (3.0–9.0 mm) sizes in the upper layer at z ; 9.0 km due to two effects: 1) the strong competition among ice crystals for the moisture there, and 2) the supersaturation, although positive, is limited to d i 5 0%–5.0% (see Fig. 2). At lower heights, the competition between crystals for the available moisture is lessened. Because the supersaturation is larger and the crystals grow faster, their mean radii reach 25–33 mm at heights of 7.0–8.0 km. Still lower, the mean radii decrease with decreasing supersaturation. In agreement with aircraft observations (see Fig. 9 of Sassen et al. 1989), the mean radii tend to increase with decreasing height and have a maximum at some height not far above the cloud bottom. Since the formation of the vertical profile of r I takes place in the whole cloud depth simultaneously and cannot be explained by gravitational sedimentation alone (i.e., crystal terminal velocities are insufficient for the crystals to fall through the whole

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cloud during this time), the fastest growth obviously occurs in those regions with maximum supersaturation. The important cloud microstructure characteristic of the ice crystal size spectra is shown in Fig. 4 (top, for 30 min) at four different horizontal locations. As one can see, they display the following common features regardless of location. At upper levels (z 5 9.0 km) the spectra have maxima r I 5 4.0–7.0 mm and rather steep slopes to larger radii. That is, the spectra are narrow because crystal growth is slowest at these levels, as discussed above. With decreasing altitude (z 5 8.7, 8.1, and 7.5 km), the maxima decrease and are displaced to values of r I 5 20–30 mm, while the slopes are much less steep as the spectra broaden due to the growth of the larger crystals. Often the spectra become bimodal due to two processes: 1) the aggregational growth of the larger crystals at the expense of small crystals and 2) the turbulent mixing between different levels. This is especially evident at x 5 33 km, where the vertical velocity, supersaturation, and crystal growth rate are at their maxima (see Fig. 2). At the lowest height of z 5 6.9 km, the peak in the size spectra is again displaced to lower sizes (5.0–7.0 mm) because the supersaturation and crystal growth rate are quite low here. Crystal evaporation begins to occur just below this level. 2) 60-MIN

SIMULATION

As Fig. 2 (right) shows, after 1 h the dynamical disturbances have not disappeared, although it is mostly the perturbations with scale lengths of ;10 km that have survived. There are several vortices in the field of c, which produce five main regions of weak updrafts with w 5 3–6 cm s21 . There are also regions of downward perturbations between the updrafts, but these are masked by the superimposed positive Wsyn with a 5.0 cm s21 maximum in the x 5 30–40-km region. The mesoscale perturbations are comparable with the large-scale velocity. Such a horizontal structure with a dominant wavelength of ;10 km again coincides with aircraft observations (Sassen et al. 1989; Smith et al. 1990). Surprisingly, however, the supersaturation field has not changed markedly after the additional 30 min of simulation, because the vapor sink onto growing crystals has been compensated for to a large extent by supersaturation production (as is described in detail below). The maximum d i has actually increased slightly in the region (x ; 35–40 km) with maximum synoptic updrafts. It should be recalled that our domain is moving with the mean wind and follows the prevailing area of large-scale updrafts created by, for example, a ULT, which serves as a persistent source of supersaturation production. As before, a vertical cross section of the cloud microstructure is shown in Fig. 2 (right side). Here N I has increased slightly to 160–200 L21 in the crystal generation layer at z ; 9.0 km, which is the coldest region where the supersaturation with respect to ice remains

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FIG. 4. Crystal size distribution functions (L21 mm21 ) after 30 min (upper row) and 60 min (lower row) of simulation at four horizontal locations (x 5 9, 33, 69, 84 km) and five heights (6.9, 8.1, 8.5, 8.7, 9.0 km), corresponding to the two-dimensional microphysical fields presented in Fig. 3.

positive, and has also increased slightly below. The maximum IWC has increased in 30 min by 3–4 times, reaching 23 mg m23 , and is located in the domain of maximum d i and Wsyn at z 5 8.0 km and x 5 20–40 km. IWC has decreased at both boundaries to 3–5 mg m23 , along with d i and Wsyn . This shows that despite the sensitive modulation of the vertical velocity and IWC fields by mesoscale updrafts, the main process of IWC accumulation is caused by the synoptic-scale velocity. Moreover, the mesoscale updrafts vary and change their positions with respect to the synoptic updraft relatively fast, while the process of deposition, determined by the crystal phase relaxation time, as described below, may take 0.5–3 h. The vertical profile of r I in Fig. 3 has now developed a more typical profile for mature cirrus: it has a minimum of 5.0–7.0 mm near cloud top and increases to a maximum of 30–50 mm near cloud base. We underscore again that the processes responsible for the vertical profile of r I proceed throughout the entire cloud simultaneously. At this stage, this is still controlled by the vertical profiles of vertical velocity and radiative cooling, although at later times gravitational settling can contribute considerably to the differentiation of crystal

sizes. In the horizontal direction, the mean radius has a maximum at x 5 20–40 km where vertical velocity and supersaturation are maximum and the crystal growth is the fastest. The crystal size distributions after 1 h (Fig. 4, lower row) again exhibit strong spatial variability. Comparison with the previous results reveals that except for the uppermost layer at z 5 9.0 km, where modal radii are again ;5–7 mm, the modal radii have increased by about a factor of 2 and reached 40–50 mm. (Note the different scales used for the top and bottom rows.) This feature can be explained mainly by depositional growth. However, the population of the largest crystals in the lower layer (z 5 6.9 km) has increased the fastest and is now comparable or exceeds the populations aloft due to gravitational sedimentation from above. The bimodality in the size spectra of this lower layer is especially pronounced due to two reasons: 1) the small crystals formed locally and growing slowly at low supersaturations have mixed with the much larger crystals falling from aloft, and 2) the oldest sedimenting crystals have been subjected to a prolonged process of gravitational accretion (aggregation), which contributes to the separation of the particle size modes. Although all the spectra have dif-

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FIG. 5. Budget of supersaturation (mg m23 h21 ) after 30 min (left) and 60 min (right) of cirrus development: generation rate due to dynamical and radiative forcing (dd i /dt)gen 5 GEN.DELI , depletion rate due to deposition on crystals dd i /dt)dep 5 DEPL.DELI , and total budget of supersaturation production BUDGET DELI .

ferent slopes that must be accounted for during the construction of models with parameterized microphysics, Fig. 4 indicates that an appropriate parameterization is the gamma distribution with parameters depending on space and time, as was done in the radiation component of our model (see Part I). We point out that similar parameterizations can be used in bulk cloud models or even in GCMs, where IWC is being calculated, or reasonably prescribed. It is noteworthy that even in developed cirrus there are numerous small ice crystals in the upper cloud region. This appears to be in good agreement with recent in situ measurements using devices especially designed for the detection of small ice particles like the CVI (Strom et al. 1994), HYVIS (Mizuno et al. 1994), VIPS (Heymsfield and Miloshevich 1995), and formvar replicators (Arnott et al. 1994). We shall show later that the abundance of these small ice crystals at the upper

cirrus levels can strongly influence the optical and radiative properties of the cloud system as a whole. b. Supersaturation budget The above discussion shows that cirrus cloud development is governed to the first order by the supersaturation field d i (x, z), with all its characteristic features and inhomogeneities. In order to understand the processes governing the evolution of the supersaturation field, we shall analyze the following equation for the local (loc) supersaturation balance [see the derivation and the discussion in Part I, Eq. (25)]. 1) 30-MIN

SIMULATION

The analysis of the supersaturation production budget is given for the 30-min time in Fig. 5 (left). The su-

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persaturation generation rate (dd i /dt)gen (denoted by GEN.DELI ) has a patchy structure similar to the vertical velocity field (Fig. 2), which is the main factor in supersaturation production. Generation is positive in the areas of updrafts with maxima of 25–30 mg m23 h21 at z ; 8.0 km and is negative in downdrafts with minima of 210 to 215 mg m23 h21 . But this field does not simply coincide with the field of w, because it is also modulated by radiative cooling, which, in maximizing at heights of 8.5–9.0 km, cools the entire layer and thus produces positive supersaturation (see below). The higher the IWC, the stronger the radiative cooling is in the upper portion of the cloud, and the enhanced supersaturation production thus increases IWC. So a positive feedback exists between (dd i /dt)gen , IWC, and radiative cooling. At the same time, depletion of supersaturation (Fig. 5, middle, labeled DEPL.DELI) takes place due to vapor deposition onto ice crystals. The maximum rate of 215 to 220 mg m23 h21 at z 5 8.0–8.5 km corresponds to the fastest crystal growth region. The total budget of supersaturation production (generation plus depletion) is shown at the bottom of Fig. 5 (BUDGET.DELI ): it has a noticeably patchy structure. The total budget is positive (supersaturation is increasing) from x 5 18 to 44 km (especially below z 5 8.0 km), is strongly negative (supersaturation is decreasing) from x 5 45 to 55 km below 8.0 km and from x 5 40 to 60 km above 8.0 km, and is again positive beyond 60 km with three distinct maxima caused by three updraft cells. However, if the typical values of supersaturation production–deposition reached after 30 min are 15–30 mg m23 h21 , one may ask why the maximum IWC is only 4–6 mg m23 . In order to answer this question, one must introduce two important quantities that characterize the noninstantaneous nature of the deposition process: the excess of uncondensed vapor (supersaturation) VAP.EXCESS, and the relative amount of the condensed ice or percent of condensed ice REL.ICE. The quantity VAP.EXCESS represents the amount of uncondensed vapor that is available for deposition and could be transformed into ice (as is usually done in bulk cloud models or GCMs) but is still uncondensed in this microphysical model, thereby accounting for the limited capability of the growing crystals to absorb the available supersaturation. REL.ICE is the ratio of the IWC condensed by the model, to the amount of ice what could be realized if an instantaneous transformation of supersaturation into IWC were to be assumed. These quantities can be written as VAP.EXCESS 5 d i q si REL.ICE 5 IWC/(IWC 1 VAP.EXCESS). The slowness of the deposition process and its consequences are seen in Fig. 6 (left). The crystal phase relaxation time t fc [see Part I, Eq. (22)] varies from a minimum value at 30 min at the center of the cloud, 60–90 min at the lateral boundaries, and 2–3 h near the

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top and bottom of the cloud. Obviously, the vapor excess is not transformed instantaneously into the condensed ice phase, and the process of deposition may actually require a period of hours. This is why VAP.EXCESS has reached 10–20 mg m23 (Fig. 6, middle), which is 3–6 times more than the modeled IWC, and REL.ICE is only 3%–30% (Fig. 6, bottom). Only this small amount of supersaturation is condensed after a half-hour despite the rather high crystal concentrations nucleated. 2) 60-MIN

SIMULATION

The effects of an additional 30-min simulation on the supersaturation budget is shown on the right side of Fig. 5. There are six areas of positive generation of supersaturation with values up to 25–40 mg m23 h21 in the layer from 7.0 to 9.0 km, stemming from positive vertical velocities (Fig. 2). Below 7.0 km, supersaturation production is almost everywhere negative, which is caused by the small negative vertical velocities and the total radiative heating, leading to the evaporation of the crystals. (As shown below, infrared cooling here is about 20.7 to 21.0 K day21 , while solar heating reaches 1.2–1.5 K day21 .) The depletion of supersaturation due to vapor absorption by crystals (Fig. 5, middle) has nearly doubled near z 5 8.0 km in 30-min because of the increased ice mass, and its capability to absorb vapor. Equation (25) from Part I shows that (dd i /dt)dep is inversely proportional to t fc , which is determined by N I and r I . With the larger mean sizes prevailing at this stage, t fc is less: 25–40 min at z 5 7.5–8.5 km in contrast to the 50–100 min calculated earlier, and the absorption rate is faster. Thus, there is a positive feedback between the amount of condensed ice mass and the rate of its absorption. On the other hand, below 6.7 km, (dd i /dt)dep . 0 and attains values of 4.2 mg m23 h21 due to the evaporation of crystals precipitating into regions where d i is negative (see Fig. 2). This results in the moistening of the subcloud air and the growth of saturation. A simple estimation of the cooling rate caused by these evaporating crystals is 20.05 K h21 , which is quite comparable to that obtained from the analysis of experimental data by Gultepe and Starr (1995), for their cases 2 and 3. The total supersaturation budget after 60 min (Fig. 5, bottom right) is negative almost everywhere in the domain above 6.8 km because the rate of crystal vapor absorption now exceeds the rate of supersaturation production. The only exception is the area near the right boundary for x . 80 km, where production is strong while IWC and absorption are small and the total budget is positive (i.e., supersaturation increases). Below 6.6 km several regions with positive budget exist because moistening from crystal evaporation exceeds the local supersaturation decrease due to the total effects of heating by radiation and subsidence. The crystal phase relaxation time after 1 h of simulation (Fig. 6, right) remains large at 25–180 min: in

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FIG. 6. Characteristics of ‘‘inertia’’ of the condensation process after 30 min (left) and 60 min (right) of cirrus development expressed as: crystal phase relaxation time t fc (min), excess of uncondensed vapor VAP.EXCESS (mg m23 ), and percent of condensed ice REL.ICE (%).

other words, the process of deposition still lags. The excess of uncondensed ice defined above reaches 15– 20 mg m23 , which is comparable to the IWC, and the percent of condensed ice is only 30%–60% in the cloud. This is an extremely important finding for the correct evaluation of cirrus optical and radiative properties and should be accounted for in models. Most of the bulk mesoscale cloud, weather forecast, and climate models condense all the excess vapor in several time steps (5– 15 min) or in a GCM characteristic time of ;1 h (e.g., Ramanathan et al. 1983). This can clearly lead to a considerable overestimation of cirrus cloud optical thickness in the visible and infrared, radiative cooling– heating, latent heat release, etc. As our detailed simulations show, the process of phase transformation in cirrus may take considerably longer than 1 h, and the unique and most reliable criterion for the completion of this process is the value of the phase relaxation time. Nonetheless, a simple recommendation can be given:

decrease the amount of ice condensed in 1 h by roughly 50%–70%. This would correspondingly decrease the optical thickness by about a factor of 2 and would influence many modeled effects and feedbacks. Of course, this is a very preliminary and crude recommendation, and numerous additional runs of the model are necessary in order to clarify the relation between condensed and uncondensed ice in cirrus, and their effects on climate and climate change modeling. Further discussion of this important finding, with comparison to available experimental and numerical results, is provided in appendix A. c. Ice mass budget The parameterization of ice mass budget (see section 3e of Part I) is quite important for use in GCMs. An interesting parameterization was proposed by Heymsfield and Donner (1990), who considered this process in a Lagrangian coordinate system ascending with the

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FIG. 7. Budget of condensed ice (mg m23 h21 ) after 30 min (left) and 60 min (right) of cirrus development: deposition–sublimation rate (dM/dt)dep 5 DEPOSIT, gravitational influx of crystals (dM/dt)gr 5 GRAV.INFLUX, and total budget of crystal mass (dM/dt)tot 5 ICE BUDGET.

mean vertical velocity, under the assumption of icesaturated humidity. Their parameterization was recommended for those GCMs that do not include a prognostic equation for IWC. With our cloud model, we shall consider the ice mass budget in an Eulerian coordinate system (in the vertical). The results should be useful for those GCMs that include a prognostic IWC equation but have coarser vertical resolution than cloud models. 1) 30-MIN

SIMULATION

Results from this simulation and analyses of fluxes show that the precipitation rate of crystals (gravitational flux directed downward) reaches a maximum of 0.5–1.0 mg m22 s21 at z 5 7.6–8.5 km and x 5 20–70 km, and then sharply decreases outside this domain. The regular flux of crystals caused by vertical velocities reach only 0.1–0.3 mg m22 s21 (upward) at z 5 8.0 km and x 5

20–40 km, where vertical velocities are at a maximum, and is much less in the rest of the domain. The turbulent flux of crystals attempts to redistribute the crystals in the vertical, that is, along the gradient of ice mass (upward above 8.4 km and downward below), but its values are negligible. Thus, ice crystal precipitation plays the major role in the vertical mass redistribution at this time. These various components of the total budget of crystal mass are shown in Fig. 7, left. One can see that (dM/ dt)tot is determined mainly by (dM/dt)dep and (dM/dt)grav, which together redistribute the condensed mass. The value of (dM/dt)grav is negative above 8 km (outflow of crystals from this layer due to precipitation), reaching a minimum of 28.0 mg m23 h21 . The term (dM/dt)dep decreases by 80%–90% in the upper cirrus layer (;9.0 km) and by 10%–50% at lower altitudes. Below 8.0 km the gravitational influx is positive with a 4.5 mg m23 h21 maximum, which is comparable to, or more than,

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the influx by deposition. Thus gravitational influx increases the mass growth by deposition by 50%–150% in the cirrus below 7.5 km. As a result, condensed crystal mass grows at the maximum rate of 15–20 mg m23 h21 at z ; 8.0 km. 2) 60-MIN

SIMULATION

The further redistribution of the condensed ice by the various fluxes is depicted in Fig. 7 (right). The gravitational flux of crystals reaches 5–10 mg m23 s21 , while the regular flux caused by upward vertical velocities mostly does not exceed 0.3–1.3 mg m23 s21 . Turbulent flux again tries to redistribute the ice mass; that is, it is directed upward above the maximum IWC (z 5 8.0– 9.0 km) and downward below 8 km, but its values are ,60.1–0.2 mg m23 s21 . The total budget of the condensed ice shows that the main ice mass accumulation takes place at around 8.0 km where the growth rate reaches 15–45 mg m23 h21 . It is mostly redistributed by precipitation. Gravitational influx clearly causes a decrease in crystal mass above 7.8 km; its minimum is 220 to 230 mg m23 s21 at z ; 8.5 km and exceeds there the depositional influx by 50%–100%: the maximum of 20–40 mg m23 s21 is located at ;7.0 km and exceeds there the depositional influx by 3–6 times. Dynamical influx caused by vertical velocity and turbulence counteracts the gravitational influx, that is, it shows a tendency to increase ice mass above 8.0 km and decrease it below, but its values are 5–10 times lower. Thus, the main factors that determine the vertical distribution of ice mass after 1 h are the depositional growth and precipitation of ice crystals. The total ice mass budget is close to zero above 8.0 km (except from x 5 20–40 km, where the precipitation is strongest), and strongly positive from z 5 6.5 to 8.0 km with a maximum at ;7.5 km, where the contributions from deposition and precipitation are comparable, or precipitation tends to dominate. To conclude this section it should be stressed that the strong influence of precipitation on the vertical redistribution of ice mass in cirrus clouds is of great importance in GCMs with high vertical resolution. 4. Optical and radiative properties Important properties of clouds, which are known to depend on their microstructure, are the absorption and scattering coefficients and their derivatives, particularly the single-scattering albedo. They influence both the radiative properties of clouds (radiative fluxes and heating–cooling rates) and their characteristics of remote sensing (optical thickness at various wavelengths, emissivity, radar reflectivity, etc.). We will discuss first the longwave properties of the simulated cirrus cloud. One of the main quantities in this regard is the mass absorption coefficient, aabs m . Numerous experiments and discussions have been devoted to the value of this co-

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efficient, and to its dependence on cloud microstructure. Unfortunately, the values of aabs m determined from various experiments are very different. Stephens et al. (1990) presented a review of these values along with the methods of measurement. According to this review and other sources, we provide the following values: 2 21 aabs (Griffith et al. 1980), 3400 cm 2 m 5 760–960 cm g 21 2 21 g or 180–680 cm g (Paltridge and Platt 1981), 230– 460 cm 2 g21 (Smith et al. 1990), 460–580 cm 2 g21 (Griffith and Cox 1977), 180–680 cm 2 g21 , and 600 cm 2 g21 (Stephens et al. 1990). According to the analysis of Smith et al. (1990), ‘‘the mass absorption coefficient should vary inversely with ice particle dimension.’’ Values of aabs m are also dependent on the height in the cloud and on the geographical region where the cloud was studied. Note that after the determination of some aabs m , this prescribed value was typically applied to the entire cloud for use in radiative calculations. Thus, it was assumed that a single value of aabs m could characterize the longwave radiative properties of the whole cloud. Another important cloud property is the optical mass scattering coefficient s scm , which determines optical thickness, single-scattering albedo, and other radiative properties in the solar spectral region. In radiative calculations, s scm has often been calculated with use of Mie theory for some prescribed size spectra, and then a singular value of s scm was applied to the entire cloud. This coefficient is also very sensitive to cloud microstructure, as was shown in particular by Stackhouse and Stephens (1991). They performed calculations of the solar and longwave radiation properties for various model atmospheres and vertically uniform clouds consisting of spherical particles, using Mie theory and size spectra measured during the FIRE IFO-I at a single cloud height. Their results differed significantly with or without the somewhat arbitrary addition of a fraction of small crystals (with radii less than 15 mm), which could not be measured by the usual particle measuring system (PMS) aircraft probes. Namely, with the addition of a small crystal population, the spectral albedo in the entire solar spectrum increased by about two times and the longwave heating increased by about three times. This situation, of an extreme sensitivity of cloud optical and infrared properties to cloud microstructure is even more dramatic because, as was shown above, cloud microphysics exhibits strong spatial and temporal variability, thereby causing a corresponding variability in sc the mass absorption aabs m and scattering s m coefficients. Nonetheless, no one (to our knowledge) has previously attempted to investigate the effects of spatially and temsc porally variable aabs m and s m , which we undertake in this study based on anomalous diffraction theory (ADT) [see Part I, Eqs. (32)–(34)]. We have calculated at 30-s intervals the longwave and shortwave mass absorption–scattering coefficients for our simulated cloud over the entire 2D domain, producing the vertical profiles (for a 60-min growth time) in Fig. 8. It is seen that the maximum value of aabs m 5

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abs FIG. 8. Vertical and horizontal profiles of the longwave mass aabs coefficients, and the optical mass s scm and m and volume absorption a y sc volume-scattering s scy coefficients, after 60-min simulation time: vertical profiles of (a) aabs , (b) s scm, (c) aabs m y , and (d) s y , and horizontal profiles of (e) aabs and (f ) s scy . y

700–800 cm 2 g21 is attained in the upper part of the cloud, where the size of the crystals is the smallest (Figs. 3 and 4). (A maximum value of 950 cm 2 g21 was found at 30 min, such that the coefficients decreased slightly with time due to ice crystal growth.) These values are rather close to those reported by Griffith et al. (1980) for the upper parts of tropical cirrus. However, our values of aabs m strongly decrease in the middle and lower parts of the simulated cirrus, where supersaturation and crystal size maximize (Figs. 2 and 3), down from 200 to 300 cm 2 g21 earlier to 100–200 cm 2 g21 at 60 min (Fig. 8). These decreasing trends in aabs m with height and time are caused by corresponding increases in ice crystal 21 mean radii. Since aabs for crystals m is proportional to r i .10–15 mm [see Part I, Eq. (34)], the depositional growth of ice leads to a decrease in the absorption coefficient with decreasing height and time corresponding abs to the increase in r 21 i . These calculated a m values are

in good agreement with the values that Smith et al. (1990) retrieved from the FIRE IFO-I measurements (230–460 cm 2 g21 ), who, along with Platt and Harshvardhan (1988), also discussed the inverse dependence of aabs m on r i. The horizontal variability in aabs m can be estimated from their vertical profiles (see Fig. 8, top left). In the upper cloud at z 5 8.7 km, the value of aabs m varies from 620 cm 2 g21 near the lateral boundaries, where r i 5 6–8 mm, to 260 cm 2 g21 at x 5 40–50 km, where r i 5 15–18 mm due to the larger vertical velocities and more intensive depositional growth. At lower heights (7.5–8.1 km), values of aabs m are much less and are more uniform in the horizontal at 150–250 cm 2 g21 , as r i 5 25–30 mm. Farther downward at z 5 6.9 km, aabs m increases again due to the smaller crystal sizes and shows strong horizontal oscillations because this lowest layer was influenced by fallstreaks from above (see Fig. 3 and 4).

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These peculiarities of the longwave absorption coefficient illustrate the extent to which considerably different values could be measured by aircraft, even during one horizontal leg of 30–60-km length, while the vertical profiles reveal that measured values could vary by 3–6 times depending on the aircraft height within the simulated cloud domain. It should not be surprising that such divergent values have been reported in various experiments: aabs m depends not only on cloud type and microstructure, but also (strongly) on aircraft location within the cloud. The vertical distribution of s scm at a wavelength of l 5 0.5 mm is similar to that for the longwave radiation (Fig. 8): it displays a sharp peak of 2000–2300 cm 2 g21 in the uppermost cloud layer and decreases to 300–500 cm 2 g21 in the middle and lower parts. The horizontal distribution of the mass scattering coefficient is also similar to that for the longwave radiation owing to the same reasons. In this regard, it is important to note a feature of the visible and infrared coefficients: despite the fact that s scm and aabs m are described by different formulas, they have similar vertical and horizontal profiles. Since these coefficients determine the cloud optical thicknesses in visible and infrared regions, t vis and t IR , respectively, it is apparent that the measurement of t vis allows the estimation of t IR , and vice versa. This analysis explains the wide variance in the longwave and optical mass absorption coefficients measured in clouds, and shows that 1) single ‘‘representative’’ values of s scm and aabs m simply do not exist, since there generally exist 3D fields of local coefficients determined by the cloud microstructure, depending on the mechanisms of crystal nucleation, and the supersaturation and vertical velocity fields, which govern crystal growth; 2) our simulation suggests that s scm and aabs m in cirrus clouds may have typical vertical profiles similar to those shown in Fig. 8, profiles that are characterized by a main maximum in both coefficients in the upper crystal generation layer due to an abundance of small ice crystals, and then decreasing downward as crystal sizes increase; and 3) a strong horizontal variability in s scm and aabs m appears typical for cirrus due to variable conditions for crystal growth. The other two important characteristics that participate explicitly in radiative transfer equations, and can also be directly measured from the lidar sounding of cirrus clouds, are the volume longwave absorption and abs volume optical scattering coefficients, aabs 3 vol 5 a m IWC and s scvol 5 s scm 3 IWC, respectively. The vertical sc profiles of aabs vol and s vol in Fig. 8 show that their behavior is distinct from that of the mass coefficients. The volume coefficients display a single pronounced peak (0.1–0.35 km21 for absorption and 0.2–0.9 km21 for scattering) near the middle of the cloud at z 5 8.2–8.4 km. This is 300–400 m higher than the maximum in IWC (see Fig. 3), because the vertical distribution of the mass scattering coefficient displays a maximum in the upper layer. Although IWC increases from 30 to 60 min by

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4–6 times, the corresponding increase in both volume coefficients during this time is only about two times, because their growth is hampered by the decrease in mass coefficients. So we see that the growth of the volume coefficients with time is much slower than the growth of IWC and that its maximum is located higher than the IWC maximum. These features have implications for the analysis of lidar measurements of the volume-scattering coefficients and for drawing conclusions on the spatial/temporal evolution of IWC from such data. Horizontal profiles of the cloud volume coefficients, which can be estimated or directly measured by various lidar techniques (Sassen et al. 1990), may also provide very useful information on the dynamical structure of the atmosphere during cirrus formation. Such profiles are included in Fig. 8. They clearly reflect the superposition of the processes of two different scales accounted for in our simulation: the synoptic- and mesoscale. Synoptic-scale processes have caused the main maxima of these coefficients to be found in the region x 5 20–40 km, where persistent Wsyn updrafts in the middle region of the cloud have led to the growth of IWC and the volume coefficients. The maximum coefficients at z 5 8.1 km occur where Wsyn has a maximum in the vertical and the most vigorous processes of supersaturation generation and crystal growth take place. This reflects the impact of ‘‘large-scale condensation,’’ as it is called in GCMs. The values of both volume coefficients decrease to the lateral boundaries and to the top and bottom of the cloud along with the value of Wsyn . Horizontal mesoscale inhomogeneities in the volume coefficients are the most pronounced in the upper (z 5 8.7 km) and lower (6.9 km) cloud regions, where the variations reach 60%–100%. This picture is very similar to lidar measurements of cirrus (e.g., Sassen et al. 1989; Sassen et al. 1990). Such inhomogeneities are relatively less significant but are still apparent in the middle of the cloud at z 5 7.5–8.1 km. These variations are related to mesoscale up/downdrafts, as we can see from the abs comparison of Figs. 2 and 8. Maxima of s scvol and avol coincide with maxima in wmax in the upper (8.7 km) and lower (6.9 km) cloud layers, where the impact of the background synoptic-scale velocity is less. However, even here there is no one-to-one correspondence between s scvol and wmax (e.g., the wmax at x 5 25 km is seen abs in the profiles of s scvol and avol at z 5 6.9 km, but not at z 5 8.7 km), because the crystal phase relaxation time of cloud ‘‘reaction’’ to mesoscale perturbations is about 25–40 min (see Fig. 6). 5. Longwave and solar radiative cooling–heating rates As previous numerical studies have generally used some fixed value of aabs m , constant for the whole cloud,

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it is worthwhile to investigate what might be the consequences of such a prescription. One of the most interesting studies of the climate sensitivity to the nonblackness of cirrus clouds was carried out in the pioneering GCM research by Ramanathan et al. (1983). Based on the aircraft measurements of Griffith et al. (1980) in tropical cirrus, and on other experimental and theoretical studies in which aabs m ranged from 500 to 1500 cm 2 g21 , they chose in their climate 2 21 model a ‘‘medium’’ aabs value for a m 5 1000 cm g ‘‘variable nonblack cirrus’’ computation. They calculated the emissivity and longwave radiation fields with this value and showed that accounting for the nonblackness of cirrus fundamentally improved the climate simulation result. On the other hand, the results of GCM simulations would likely differ if it were possible to account for the vertical inhomogeneities in aabs m , such as is shown in Fig. 8. In order to investigate the possible impact of variable fields of aabs m on the longwave radiation field, we have performed a comparison of radiative profiles calculated by two methods: method A uses model-calculated profiles of aabs m and method B assumes a prescribed value of aabs m independent of microstructural variations. If an aircraft measurement of aabs m was performed near our simulated cloud top at 9.5 km, for example, it would give a value of 800–950 cm 2 g21 according to Fig. 8. This is not too different from the value of 1000 cm 2 g21 used by Ramanathan et al. (1983), and so their value will be employed in our method B as a proxy for measured values. We can then estimate what the consequences of the use of this value would be if it were to be extrapolated to the entire cloud, as we now show in Fig. 9. It is obvious that there are major differences exhibited by the two methods. Both upward and downward fluxes (Figs. 9a,b) change much less in the cloud model results with calculated aabs m than with the prescribed coefficient. Outgoing longwave radiation is increased by 40–50 W m22 using method B as compared to method A, meaning that the use of such a prescribed aabs m strongly overestimates the greenhouse effect of cirrus clouds. Even more dramatic effects are seen in the profiles of the longwave radiative temperature change (Figs. 9c,d). The cooling rates (]T/]t)long from method A exhibit cooling everywhere in the cloud layer, ranging from 20.5 to 22 K day21 in the lower cloud layer to 22.5 to 24 K day21 at z 5 8.4 km (;1.0 km below cloud top). The profile at x 5 33 km (Fig. 9c) with maximum IWC indicates that increasing the IWC more strongly cools the upper cloud and more weakly cools the lower cirrus. This shows the tendency for heating to occur above the cloud base, but the IWC there is still insufficient for heating to take place. Similar profiles, but calculated with method B (Fig. 9d), are noticeably different: they exhibit stronger heating in the lower part of the cloud below ;8.0 km and much stronger cooling aloft. Such

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profiles of infrared cooling would lead in cloud models and GCMs to a much more unstable atmosphere. This last picture (i.e., of cooling above and heating below) is considered to be the classic one in both the midlatitudes and Tropics, and it has often been reproduced in mesoscale and climate models. But our results show that, at least for this simulated cloud, it may not always be justified: it is simply a result of the constancy assumed in mass absorption coefficient with height. Since the vertical distribution of the cooling–heating rates are extremely important for weather and climate studies, the height-invariant prescription of aabs m without account for microstructural variations can overestimate the static instability and strength of convective activity in the upper troposphere. For example, Wetherald and Manabe (1988) analyzed with the use of a GCM the sensitivity of climate to the global warming due to the doubling of CO 2 . They found a positive cloud-warming feedback from the increased amount and height of cirrus near the tropopause and indicated that this could be explained by the decrease in the static stability of the upper troposphere caused by greenhouse gas increases, pointing out that ‘‘the vertical velocity depends strongly upon the static stability and tends to become weaker with increasing stability.’’ However, Fig. 9 suggests that if the radiative properties of cirrus were treated more realistically, the upper troposphere would be more stable and this would hamper the vertical development of the tropopause and cirrus during global warming, and the magnitude (and perhaps even the sign) of the cloudwarming feedback might be changed. Clearly, additional GCM sensitivity studies are necessary to clarify the range of likely cloud feedbacks accompanying global warming. Finally, vertical profiles of the solar radiation field are given in Fig. 10. The presence of the horizontal variations in cloud optical properties causes similar inhomogeneities in the solar radiative properties. The main disturbance of the upward flux takes place above 8.0 km, and the main disturbance of the downward flux lies below this height. The downward flux decreases by 170 W m22 within the cloud layer at x 5 33 km at the maximum optical thickness, but only by 70 W m22 at x 5 84 km at the smallest optical thickness (Fig. 10a). The upward solar flux increases by 140 W m22 at x 5 33 km, but only by 45 W m22 at x 5 84 km (Fig. 10b). As shown in Fig. 10c, the maximum solar heating rate reaches 3.5 K day21 at z 5 8.0 km and x 5 33 km, and decreases to 1.0–1.5 K day21 at the lateral boundaries and the cloud bottom. The albedo of the cloud and underlying surface (Fig. 10d) has a 26% maximum in the same region x 5 33 km and decreases toward the boundaries to 14%–20%. The albedo of the cloud alone in the two regions is 16% and 4%–10%, respectively. Since the solar heating exceeds the infrared cooling rate (Fig. 9) below ;8.0 km, the total radiative heating is positive and this prevents crystal growth, causing evaporation to occur.

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FIG. 9. Comparison of the vertical profiles of longwave radiative characteristics after 60 min with calculated abs 2 21 absorption coefficient aabs : (a) downward flux F ↓l at two horizontal locations (W m vs prescribed a m 5 1000 cm g m22 ), (b) upward radiative flux F ↑l (W m22 ), (c) radiative heating–cooling rate (]T/]t) l (K day21 ) at four vertical 2 21 locations with calculated a, and (d) the corresponding heating–cooling rates with prescribed aabs . m 5 1000 cm g

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FIG. 10. Vertical profiles of solar radiation at various horizontal locations after 60 min of cirrus development: (a) downward flux F ↓s (W m22 ), (b) upward flux F ↑s (W m22 ), (c) solar radiative heating rate (]T/]t) s (K day21 ), and (d) albedo A(z) (%).

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FIG. 11. Horizontal profiles of the radiative balance components (W m22 ) above the cloud after 60 min: longwave balance R l , shortwave balance R s , total balance Rtot , and the horizontal profile of albedo (%).

6. Radiative balance One of the most important quantities for climate considerations is the radiative balance of cirrus clouds or the net radiative flux that we define to be positive downward; that is, the longwave balance is R l 5 F ↓l 2 F ↑l , the shortwave balance is R s 5 F ↓s 2 F ↑s , and the total balance is R t 5 R l 1 R s. As the perturbations of the radiative fluxes at the right boundary, x 5 x r , with a thin cloud are relatively small, the horizontal variations of dR l (x) 5 R l (x) 2 R l (x r ) above the cloud characterize longwave cloud forcing, and similar variations dR s , dR t , represent SW and total cloud forcing, respectively. Their values characterize the albedo-versus-greenhouse effect of the cloud. The horizontal distributions of the radiative balance components for 60 min are shown in Fig. 11. Impacts of both synoptic-scale and mesoscale dynamical forcing are clearly seen in the horizontal distributions of the radiative balance components and albedo. At the top of the domain, maximum R lmin 5 2153 W m22 and minimum R lmax 5 2182 W m22 , such that the maximum decrease in dR l (a greenhouse effect or longwave cloud forcing) is 29 W m22 at x 5 35 km. Secondary maxima are seen in the areas of mesoscale updrafts, especially at x 5 7, 50, 70, and 85 km. Maximum and minimum values of shortwave radiative balance above the cloud are R smax 5 585 W m22 and R smin 5 512 W m22 , and the decrease in the shortwave balance (SW cloud forcing) is dR s 5 273 W m22 (Fig. 11b). Thus, the total cloud forcing is negative, dR t 5 244 W m22 . Corresponding values of the shortwave balance beneath the cloud are: R smax 5 565 W m22 , R smin

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5 480 W m22 , and dR s 5 285 W m22 . This shows that the maximum decrease in the shortwave balance within the cloud is 12 W m22 , which is much more than the decrease in the longwave balance. Thus the albedo effect of the cloud at this time (local noon) exceeds the greenhouse effect, that is, overall the albedo effect dominates. Note that earlier or later (with lower sun) the effect might be different, but around local noon, this simulated cirrus cloud causes a pronounced cooling effect. The horizontal profile of albedo as it would be measured by an aircraft above the cloud (Fig. 11) has maxima that coincide with the vertical velocity maxima. This clearly reflects the dynamic structure of the atmosphere and shows that these radiative horizontal profiles can serve not only for the estimation of cloud optical properties, but also for the study of atmospheric dynamics. Both synoptic-scale and mesoscale updrafts can be determined from the local maxima of albedo at all altitudes, in agreement with the experimental data and conclusions of Smith et al. (1990), who retrieved the turbulent structure from the radiative measurements. Finally, these results can be related directly to cloud characteristics that are usually used in GCMs and some cloud models: the longwave optical (absorption) thickness t IR , visible optical thickness t VIS , and emissivity «, which strongly depend on the spectral band. Here, we consider two spectral regions that make a major contribution to the radiative cooling–heating of cirrus in the longwave: left and right portions of the 8–12-mm atmospheric window, which, due to their transparency, are used in AVHRR channels 4 and 5 (e.g., Brogniez et al. 1995). Values of t IR were evaluated for these channels using the imaginary part of the ice refractive index k l4 5 0.177 and k l5 5 0.407. As earlier, we calculated t IR through the use of method A (aabs m calculated from the microstructure) and method B (using the prescribed 2 21 aabs ). The computed horizontal profiles m 5 1000 cm g of t VIS , t IR , and their ratio are shown in Fig. 12a. Values of t VIS reach 1.3 at x 5 25–45 km, and decrease by 2– 6 times at the cloud boundaries. Infrared t IR obtained with method A peak at 0.52 in the same region and decrease to 0.1–0.2 at the boundaries. In contrast, t IR (1000) are 3–5 times greater. Once again this underscores the necessity of accounting for the vertical cirrus structure and the typically strong downward decrease within the cloud. Note that there is no simple relation between visible and infrared optical thicknesses. The expression Rtau 5 t VIS /t IR 5 2 would be valid only if particles large relative to the wavelength were exclusively present and the Mie extinction (ext) and absorption (abs) efficiency factors could be related by Qext 5 2 and Qabs 5 1 in the large particle limit. In our case this simple relation is not valid due to the abundance of small ice particles in the upper layer. Actually, our value of Rtau ø 2.5 is very similar to some FIRE IFO-I cases (Minnis et al. 1990).

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FIG. 12. Characteristics of remote sensing after 60 min: (a) visible optical thickness t vis , infrared optical thicknesses t IR (obtained with calculated aabs m ) and t IR (1000) (obtained with prescribed 2 21 aabs ), and horizontal profile of t vis /t IR ; (b) emissivities for Advanced Very High m 5 1000 cm g abs 2 Resolution Radiometer channels 4 and 5 obtained with calculated aabs m , and with a m 5 1000 cm g21 (channel 5).

Again, optical thicknesses and emissivities clearly reflect the dynamic structure of the atmosphere and the mesostructure of the cloud. These peculiarities in aabs m and t IR strongly influence the longwave emissivity (Fig. 12b). Emissivities « 4 and « 5 for the Advanced Very High Resolution Radiometer channels 4 and 5 obtained from method A are very similar and vary from 0.1 to 0.4, which are far from black. Emissivity for channel 5 calculated with method B differs significantly from method A and reaches a maximum of « 5 5 0.97 at x 5 35 km. For all horizontal locations the method B emissivities are much more black than those calculated using the model predictions. This is important both for the remote sensing of cirrus clouds and for radiative calculations of the longwave properties of cirrus in GCMs. Note that the ice water path (IWP) of our cloud can be characterized as moderate: IWP 5 36 g m22 at x 5 33 km and decreases to 16 g m22 and 3.5 g m22 at the boundaries. This is comparable to the cirrus cloud IWP produced in a GCM by Ramanathan et al. (1983), where their lower IWP limit was 25 g m22 . Based on these results, a useful recommendation can be suggested for the evaluation of emissivity in climate models, which is described in appendix B. 7. Summary and conclusions A 2D version of a 2D/3D cloud model complex with explicit microphysics and radiation, previously used for orographic and other cloud types, has been applied here

to the numerical simulation of cirrus cloud development and its interaction with radiation. We would like to emphasize that this simulation is presumed appropriate for an occurrence of developing cirrus clouds generated in a stable atmosphere by the relatively slow, synopticscale ascent present within an upper-level shortwave trough; radiatively, the conditions correspond to midlatitude autumn over a water surface, with the 1-h simulation beginning at local solar noon. The main results illustrate the following poorly understood properties of such a rather typical midlatitude cirrus cloud. 1) The process of vapor deposition to ice crystals in cirrus is far from instantaneous. Rather, it is determined by the crystal phase relaxation time, which represents the characteristic time of supersaturation absorption by crystals in the absence of further water vapor generation. This time varies in our simulation from 0.5 to 3 h. Even after 1 h of cloud development, there can be a residual level of vapor supersaturation with respect to ice on the order of 10%, while the percent of condensed ice is only about 40%–60% of the supersaturation that could potentially be converted to ice during this time. In other words, the amount of condensed ice is comparable to the amount of uncondensed excess vapor. Since many of the bulk cloud models and GCMs transform the entire vapor excess in a few time steps (i.e., 5–10 min), or in a model characteristic time of about 1 h, this procedure can overestimate by a factor of two or more the IWC, optical thickness, emissivity, latent heat release, etc. The treatment of condensation in cloud and climate models should be revised where it is unrealistically too rapid.

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We have offered a simple, preliminary recommendation for use in GCM testing: decrease the amount of ice condensed in 1 h by a factor of 2. Experimental confirmation of the possible high residual equilibrium supersaturation is needed either in laboratory experiments or in situ measurements. 2) Numerous aircraft experiments and theoretical investigations have produced quite different values of the longwave mass absorption coefficient aabs m , ranging from about 100 to 3500 cm 2 g21 . Our results obtained with an interactive microphysical–radiative model indicate that a single representative value of aabs m simply does not exist. There is generally a 3D field of aabs m that depends on the precise cloud microstructure and so determines the fields of the optical thickness, emissivity, fluxes, and cooling–heating rates. Similar 3D fields of the s scm scattering coefficient determine the optical properties in the solar spectrum. However, according to this study, the coefficients appear to have typical vertical profiles that display a maximum in the upper (0.5–1.0 km) part of the cloud containing the smallest crystals of aabs m 5 800– 1000 cm 2 g21 and s scm 5 2500–3000 cm 2 g21 , with much lower values in the main cloud containing larger crystals 2 21 of aabs and s scm 5 300–400 cm 2 m 5 100–200 cm g g21 . Parameterization of these profiles as given in appendix B might be used for improving algorithms of cirrus remote sensing both from the ground and from satellites. 3) Many GCM and climate models rely on seemingly unrepresentative values for these coefficients (e.g., 2 21 aabs ), which are several times greater m 5 1000 cm g than seem to be appropriate for a typical cirrus. This overestimation may be extremely important for climate and climate change studies. Sensitivity tests performed here have shown that this can lead to a significant alteration of the vertical distribution of cooling–heating rates, in that the cooling in the upper part of the cloud and the heating in the lower part are both strongly enhanced. Such approaches can cause an unjustifiable increase in the static instability and cirrus cloud convective activity in the upper troposphere. It follows that a more correct accounting in models for the dependence of these coefficients on cloud microstructure would create a much more stable atmosphere and would influence cloud-radiation feedbacks. In particular, the postulated effects of the positive feedbacks between clouds and global warming induced by greenhouse gases might be different in magnitude (if not in sign) if it were possible to account for the dependence of the absorption, and scattering coefficients on cloud microstructure, as recommended here. 4) The decrease in the shortwave radiative balance (albedo effect) for our simulated cirrus cloud exceeds the net gain in longwave balance (greenhouse effect) for the simulation performed near local noon. Thus, the albedo effect in this case dominates primarily as a result of the abundance of small crystals in the upper cloud regions. This result might be verified using satellite ob-

servations, and in particular, ISCCP data (Rossow and Schiffer 1991). It is acknowledged that these results are preliminary and would likely differ somewhat under conditions other than those assumed here: for example, the initial state of the atmosphere (static stability, relative humidity, temperature, winds, and the state of the underlying surface), synoptic situation, mechanism of crystal nucleation, cloud altitude, time of day, etc. Many more microphysical model runs and comparisons with experimental findings are necessary to lessen the current uncertainties and make final recommendations for GCM parameterizations, and thus clarify the effects of cirrus cloud microphysical and radiative properties on the earth–atmosphere climate system. We believe that this can be facilitated within the framework of the GEWEX Cloud System Study and will in the future contribute to WG-2 cirrus cloud simulation research. Nonetheless, the use of explicit microphysics and radiation in the initial tests of this model have provided new insights into cirrus cloud properties, suggesting that revisions to current modeling approaches are, in many cases, in order. Acknowledgments. This research has been partially supported by Grant DE-FG03-94ER61747 from the Department of Energy Atmospheric Radiation Measurement program, and by NSF Grant ATM-9528287. The authors thank the three anonymous reviewers for their insightful remarks. APPENDIX A

Crystal Phase Relaxation Times and Equilibrium Supersaturations After the primary relaxation of supersaturation has passed, a quasi-equilibrium state may be reached, when dd/dt ø 0. We can then determine from Eq. (25) in Part I the equilibrium value of deq i and the value of the relative supersaturation

deq i,rel 5 C T(g a w 2 (]T/]t)rad )t fc ,

(A1)

where C T 5 (1/q s )(]q s /]T), q s the saturation ratio, and t fc 5 (4pDN Ir I )21 is the crystal phase relaxation time for crystals with concentration N I and mean equivalent radius r I . The term C T varies weakly with temperature, but for cirrus applications we can write as an approximation: 23 deq wt fc%, i,rel 5 1.1 3 10

(A2)

where vertical velocity w has units of cm s21 , and t fc of s. As we have seen, values of relaxation time and supersaturation strongly depend on the crystal number density and their mean size. In order to understand how general are the results described in this paper and to allow us to intercompare with other simulations and

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VOLUME 55

TABLE A1. Crystal phase relaxation times (top line) and IWC (bottom line, g/m 23) for various crystal concentrations and mean radii. Crystal concentration Nc (L21) Radius

1

5

10

100

200

500

1000

3000

1 mm

1000 h 4.0E-9 500 h 3.0E-8 200 h 5.0E-7 100 h 4.0E-6 50 h 3.0E-5 20 h 5.0E-4 10 h 4.0E-3 5h 3.0E-2 2h 4.7E-1 1h 3.8

200 h 2.0E-8 100 h 1.5E-7 40 h 2.4E-6 20 h 1.9E-5 10 h 1.5E-4 4h 2.4E-3 2h 1.9E-2 1h 1.5E-1 24 min 2.4 12 min 19

100 h 3.8E-8 50 h 3.0E-7 20 h 4.7E-6 10 h 3.8E-5 5h 3.0 E-4 2h 4.7E-3 1h 3.8E-2 30 min 3.0E-1 12 min 4.7 6 min 38

10 h 3.8E-7 5h 3.0E-6 2h 4.7E-5 1h 3.8E-4 30 min** 3.0E-3 12 min 4.7E-2 6 min 3.8E-1 3 min 3.0 72 s 47 36 s 380

5h 7.5E-7 2.5 h 6.0E-6 1h 9.4E-5 30 min 7.5E-4 15 min 6.0E-3 6 min 9.4E-2 3 min 7.5E-1 90 s 6.0 36 s 94 18 s 750

2h 1.9E-6 1h 1.5E-5 24 min 2.4E-4 12 min 1.9E-3 6 min 1.5E-2 144 s 2.4E-1 72 s 1.9 36 s 15 14 s 240 7.2 s 1900

1h 3.8E-6 30 min 3.0E-5 12 min 4.7E-4 6 min 3.8E-3 3 min 3.0E-2 72 s 4.7E-1 36 s 3.8 18 s 30 3.6 s 470 3.6 s 3800

20 min 1.1E-5 10 min 9.0E-5 4 min 1.4E-3 3 min 1.1E-2 1 min* 9.0E-2 24 s 1.4 12 s 11 6s 90 2.5 s 1400 1.1 s 11 000

2 mm 5 mm 10 mm 20 mm 50 mm 100 mm 200 mm 500 mm 1000 mm

* Similar to Fig. 4 from Sassen and Dodd (1988), with w 5 1 m s21, Nc 5 3 3 103 L21, rc 5 20 mm, and Heymsfield and Sabin (1989), at 1 min. ** Similar to Fig. 4 of DeMott et al. (1994), with w 5 5 cm s21, Nc 5 100L1, rc 5 20–30 mm, at 30 min.

measurements, we have calculated the values of t fc and equilibrium deq i,rel for a large number of situations. The results for t fc are presented in Table A1 (along with IWC for reference), and the corresponding values of deq i,rel are presented in Table A2, along with the excess of uncondensed vapor at T 5 2408C, for high, medium, and low crystal number densities.

a. Example 1: High crystal concentration (NI 5 2000–3000 L21 ) This extreme case of minimum t fc (right column in Table A1) was simulated by Sassen and Dodd (1989) and Heymsfield and Sabin (1989) with parcel models and corresponds to quite high values of N I in strong

TABLE A2. Equilibrium supersaturation with respect to ice (%, top row), and excess of uncondensed ice at 2408C (mg m23, bottom row). Phase relax. time 10 s 60 s 100 s 500 s 1000 s 30 min 60 min 120 min 180 min * See Table A1. ** See Table A1.

Vertical velocity (cm s) 1 0.01 0.01 0.04 0.05 0.07 0.08 0.35 0.39 0.70 0.78 1.26 1.40 2.52 2.80 5.04 5.59 7.56 8.39

5 0.04 0.04 0.21 0.23 0.35 0.39 1.75 1.94 3.50 3.88 6.30** 6.99** 12.6 134.0 25.2 28.0 37.8 42.0

10

20

50

100

150

0.07 0.08 0.42 0.47 0.70 0.78 3.50 3.88 7.00 7.77 12.6 14.0 25.2 28.0 50.4 55.9 75.6 83.9

0.14 0.16 0.84 0.93 1.40 1.55 7.00 7.77 14.0 15.5 25.2 28.0 50.4 55.9 100.8 111.9 151.2 167.8

0.35 0.39 2.10 2.33 3.50 3.88 17.5 19.4 35.0 38.8 63.0 69.9 126.0 139.9 252.0 279.7 378.0 419.6

0.70 0.78 4.20* 4.66* 7.00 7.77 35.0 38.9 70.0 77.7 126.0 139.9 252.0 279.7 504.0 559.4 756.0 839.1

1.05 1.17 6.30 6.99 10.5 11.7 52.5 58.3 105.0 116.5 189.0 209.8 378.0 419.6 756.0 839.2 1134 1259

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updrafts of 1 m s21 under neutral or unstable stratification (i.e., embedded cirrus convective cells). Experimental results from Sassen and Dodd (1988) show that the initial rapid crystal growth up to ;20 mm was favored by the existence of highly supercooled water (via the Bergeron–Findeisen mechanism) at cloud base, with the cloud glaciation taking place within the lowest 50– 100 m. (During this time, humidity was close to water saturation, amounting to an ice supersaturation of 35%– 40%.) Following the phase change at a height of ;100 m, N I 5 3000 L21 , IWC 5 0.2 g kg21 (;0.1 g m23 ), and r i ù 20 mm. We see from Table A1 that t fc 5 1 min, and according to our previous discussion, deq i,rel should decrease by e-times during the following 60 s. The microphysical model results given in Fig. 4 of Sassen and Dodd (1988) show that the real decrease of deq i,rel is very close to that described by the value of t fc 5 1 min. Equilibrium supersaturation for this case (Table A2) is 4.2% for t fc 5 60 s, or about 3% for t fc 5 40 s. Final modeled values of 2%–4% are obtained in Sassen and Dodd (1988) at 200-m height, as equilibrium is approached; since their results are consistent with those described here, the concepts of phase relaxation time and equilibrium supersaturation are seen to be useful for describing crystal growth, even for relatively strong cirrus updrafts. b. Example 2: Medium crystal concentration (NI 5 100–200 L21 ) This case (middle of Table A1) is probably typical for weak vertical velocities near the tops of cirrostratus clouds and is consistent with the results in this paper. In DeMott et al. (1994), the processes of crystal nucleation and growth were simulated with a parcel model for a number of vertical velocities. Using their results, we can estimate phase relaxation times and the rates of supersaturation depletion. In particular, for the w 5 5 cm s21 case, which is similar to that described here, their Fig. 4 shows that the stage of nucleation was completed at T 5 2438C, with the final value N I 5 100 L21 , and r I ù 20 mm: thus we estimate t fc as ;30 min. During the following ascent from 308 to 307 mb, which corresponds to an ascent of ;25 m, d i,rel decreased from ;40% to 28%. An estimation using t fc 5 30 min shows that during the corresponding 500 s, d i,rel would decrease to 30%, which is very close to their modeled value. The most remarkable fact here is that even 10 min after the formation of ice crystals, supersaturation is still about 30%, and 75% of all available vapor remains uncondensed. We can also estimate that supersaturation in their parcel 30 min after homogeneous nucleation should be ;15%, and similar estimates can be obtained for the case when both homogeneous and heterogeneous nucleation modes occurs. These findings are quite comparable with our supersaturation results obtained with similar vertical velocities.

The value of equilibrium supersaturation for w 5 5 cm s21 , N I 5 100 L21 , and t fc 5 30 min is deq i,rel 5 6% (see Table A2). However, for the same t fc the value of 21 deq , and 25% for w i,rel reaches 12% for w 5 10 cm s 5 20 cm s21 . Since this is still insufficient to trigger a new event of homogeneous nucleation, the process of heterogeneous nucleation might be important at this stage. c. Example 3: Extremely low crystal concentration (NI 5 1–3 L21 ) These values are probably typical near cloud bottom after primary nucleation, as can be seen, for example, from present simulation, the mesoscale bulk model by DeMott et al. (1994), and the 1D microphysical of model of Jensen et al. (1994). Table A1 shows that the corresponding crystal relaxation times are of the order 50– 1000 h, such that the process of deposition is extremely slow under these conditions. APPENDIX B

Absorption Coefficient and Cloud Emissivity The cloud emissivity can be written in two different forms that correspond to two methods considered here: method A, which accounts for the vertical profile of absorption coefficient

5

E

6

zb

« i 5 1 2 exp 2

a mabs r i (z)q Li (z) dz ,

zt

(B1)

and method B currently used in most GCMs, many cloud models, and in remote sensing, which assumes some averaged constant in height aabs m and operates directly with IWP:

[

« i 5 1 2 exp 2a abs m

E

zb

]

qLi (z) dz

zt

5 1 2 exp(2a abs m IWP).

(B2)

The results presented above (Fig. 12b) show that the use of Eq. (B2) may lead to substantial errors. Thus we can recommend for calculations of cirrus radiative properties the use of Eq. (B1) with the vertical profiles of the mass absorption coefficient as shown in Fig. 8. These profiles can be related to the profile of mean crystal radius r 1 (z) or effective radius reff (z) either with use of Eq. (33) from Part I for the mass absorption coefficient or by the parameterization by Ebert and Curry (1992) [Eq. (35) in Part I]. Using the experimental vertical structure of developed midlatitude cirrus from Sassen et al. (1989), and results of modeling described here, we can parameterize the vertical profiles of mean radius and IWC similarly as parabolic profiles:

1844


1 21 2 z z r (z) 5 c r 1H2 11 2 H2 a1b a1b C 51 , a 2 1 b 2

IWC(z) 5 c N IWCmax

z H

a

12

a

I

N

b

b

I,max

a

z H

(B3)

b

N

z max 5

a H, a1b

(B4)

where H 5 z b 2 z t is cloud depth, z b and z t the lower and upper boundaries, values with index ‘‘max’’ are the maximum (in vertical) values, a and b are the parameters that determine the location of the maximum in the vertical, and C N is a normalizing constant. For parameterizations (B3), (B4), maximum values are located at relative height z 5 zmax /H and zmax is the location of the maxima determined from the condition (d IWC/dz) 5 0. For the developed cirrus we can choose a 5 1/4, b 5 1, then maxima are located at zmax 5 0.2 H in the lower quarter of the cloud, in agreement with measurements from Sassen et al. (1989) and our modeling results, then C N 5 1.87. If one assumes a maximum (in height) radius rmax 5 50 mm (which is reached at the same height near cloud bottom), then the crystal radius near cloud top is 5–10 mm in agreement with present modeling and numerous recent measurements. Special tests showed that the parameterization of the mean radius provide vertical profiles of the mass absorption coefficient similar to that shown in Fig. 8. Some cirrus clouds exhibit maximum IWC in the middle part of the layer (e.g., tropical anvils; Griffith et al. 1980). In this case, we can choose in (B3), (B4) parameters a 5 1, b 5 1, then zmax 5 0.5 H, C N 5 4. Note that in both cases, maximum IWC ø 2/3 IWCmax . When maximum IWC and/or mean radius is located in the upper cloud part, it is possible to choose a 5 1, b 5 1/4. Parameterizations (B3), (B4), along with Eq. (B1), can be recommended for GCMs. REFERENCES Arnott, P. W., Y. Y. Dong, J. Hallett, and M. R. Poellot, 1994: Role of small ice crystals in radiative properties of cirrus: A case study, FIRE-II, November 22, 1991. J. Geophys. Res., 99, 1371– 1381. Brogniez, G., J. C. Buriez, V. Giraud, F. Parol, and C. Vanbauce, 1995: Determination of effective emittance and a radiatively equivalent microphysical model of cirrus from ground-based and satellite observations during the International Cirrus Experiment: The 18 October 1989 case study. Mon. Wea. Rev., 123, 1025– 1036. Browning, K. A., Ed., 1994: GEWEX Cloud System Study (GCSS) Science Plan. IGPO, 62 pp. DeMott, P. J., M. P. Meyers, and W. R. Cotton, 1994: Parameterization and impact of ice initiation processes relevant to numerical model simulation of cirrus clouds. J. Atmos. Sci., 51, 77–90.

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Ebert, E. E., and J. A. Curry, 1992: A parameterization of ice cloud optical properties for climate models. J. Geophys. Res., 97, 3831–3836. Griffith, K., and S. K. Cox, 1977: Infrared radiative properties of tropical cirrus clouds inferred from broadband properties. CSU Atmos. Sci. Paper No. 269, Colorado State University, 102 pp. , , and R. G. Knollenberg, 1980: Infrared radiative properties of tropical cirrus clouds inferred from aircraft measurements. J. Atmos. Sci., 37, 1077–1087. Gultepe, I., and D. O’C. Starr, 1995: A comparison of evaporative cooling with infrared heating beneath a cirrus cloud. Atmos. Res., 35, 217–232. Heckman, S. T., and W. R. Cotton, 1993: Mesoscale numerical simulation of cirrus clouds—FIRE case study and sensitivity analysis. Mon. Wea. Rev., 121, 2264–2284. Heymsfield, A. J., and R. M. Sabin, 1989: Cirrus crystal nucleation by homogeneous freezing of solution droplets. J. Atmos. Sci., 46, 2252–2264. , and L. J. Donner, 1990: A scheme for parameterizing ice-cloud water content in general circulation models. J. Atmos. Sci., 47, 1865–1877. , and L. M. Miloshevich, 1995: Relative humidity and temperature influences on cirrus formation and evolution: Observations from wave clouds and FIRE-II. J. Atmos. Sci., 52, 4302–4326. Holton, J. R., 1979: An Introduction to Dynamic Meteorology. Academic Press, 391 pp. Jensen, E. J., O. B. Toon, D. L. Westphal, S. Kinne, and A. J. Heymsfield, 1994: Microphysical modeling of circus, 1. Comparison with 1986 FIRE IFO measurements. J. Geophys. Res., 99, 10 421–10 442. Khvorostyanov, V. I., 1995: Mesoscale processes of cloud formation, cloud-radiation interaction and their modeling with explicit cloud microphysics. Atmos. Res., 39, 1–67. , and K. Sassen, 1998: Cirrus cloud simulation using explicit microphysics and radiation. Part I: Model description. J. Atmos. Sci., 55, 1808–1821. , S. Bakan, and H. Grassl, 1996. Numerical modeling contrail formation and modulation of radiation field using a mesoscale model with explicit microphysics. Preprints, 12th Int. Conf. Clouds and Precipitation, Zurich, Switzerland, Int. Commission on Clouds and Precipitation, 826–830. Minnis, P., D. F. Young, K. Sassen, J. M. Alvarez, and C. J. Grund, 1990: The 27–28 October FIRE IFO cirrus case study: Cirrus parameters relationships derived from satellite and lidar data. Mon. Wea. Rev., 118, 2402–2425. Mizuno, H., T. Matsuo, M. Murakami, and Y. Yamada, 1994: Microstructure of cirrus clouds observed by HYVIS. Atmos. Res., 32, 115–124. Paltridge, G. W., and C. M. R. Platt, 1981. Aircraft measurements of solar and IR radiation and the microphysics of cirrus clouds. Quart. J. Roy. Meteor. Soc., 107, 367–380. Platt, C. M. R., and Harshvardhan, 1988: Temperature dependence of cirrus extinction: Implications for climate feedback. J. Geophys. Res., 93, 11 051–11 058. Ramanathan, V., E. J. Pitcher, R. C. Malone, and M. L. Blackmon, 1983: The response of a spectral general circulation model to refinements in radiative properties. J. Atmos. Sci., 40, 605–630. Rossow, W. B., and R. A. Schiffer, 1991: ISCCP cloud data products. Bull. Amer. Meteor. Soc., 72, 2–20. Sassen, K., and G. C. Dodd, 1988: Homogeneous nucleation rate for highly supercooled cirrus cloud droplets. J. Atmos. Sci., 45, 1357–1369. , and , 1989: Haze particle nucleation simulation in cirrus clouds, and application for numerical and lidar studies. J. Atmos. Sci., 46, 3005–3014. , D. O’C. Starr, and T. Uttal, 1989: Mesoscale and microscale structure of cirrus clouds: Three case studies. J. Atmos. Sci., 46, 371–396. , C. J. Grund, J. D. Spinhirne, M. Hardesty, and J. M. Alvarez, 1990: The 27–28 October 1986 FIRE cirrus case study: A five

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lidar overview of cloud structure and evolution. Mon. Wea. Rev., 118, 2288–2312. Smith, W. L., P. F. Hein, and S. K. Cox, 1990: The 27–28 October 1986 FIRE cirrus case study: In situ observations of radiation and dynamic properties of a cirrus cloud layer. Mon. Wea. Rev., 118, 2389–2401. Stackhouse, P. W., and G. L. Stephens, 1991: A theoretical and observational study of the radiative properties of cirrus: Results from FIRE 1986. J. Atmos. Sci., 48, 2044–2059. Starr, D. O’C., and S. K. Cox, 1985a: Cirrus clouds. Part I: A cirrus cloud model. J. Atmos. Sci., 42, 2663–2681. , and , 1985b: Cirrus clouds. Part II: Numerical experiment on the formation and maintenance of cirrus. J. Atmos. Sci., 42, 2682–2694.

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, and D. Wylie, 1990: The 27–28 October 1986 FIRE cirrus case study: Meteorology and cloud fields. Mon. Wea. Rev., 118, 2259–2287. Stephens, G. L., S. C. Tsay, P. W. Stackhouse, and P. J. Flatau, 1990: The relevance of the microphysical and radiative properties of cirrus clouds to climate and climatic feedback. J. Atmos. Sci., 47, 1742–1753. Strom, J. R., J. Heintzenberg, K. J. Noone, J. A. Ogren, F. Albers, and M. Quante, 1994: Small crystals in cirriform clouds: A case study of residue size distribution, cloud water content and related cloud properties. Atmos. Res., 32, 125–141. Wetherald, R. T., and S. Manabe, 1988: Cloud feedback processes in a general circulation model. J. Atmos. Sci., 45, 1397–1415.

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