a. The Sierra plume

A number of interesting cloud characteristics are evident in the Sierra plume case, as illustrated by the close-up MODIS image shown in Fig. 1b. Superimposed on the main plume, there are a series of five enhanced (bright) banners, which are oriented along the plume with a cross-plume width of the order of 10 km, likely corresponding to the high mountain peaks underneath (from north to south): Mt. Morgan (3190 m), Mt. Humphreys (4263 m), Mt. Goddard (4144 m), Mt. Pelisade (4341 m), and Mt. Whitney (4418 m). The bright banners, presumably corresponding to dense clouds, decay rapidly over approximately the first 50 km. The height of the cloud plume can be estimated from the well-defined cloud shadow on the surface using H = d × tan(α), where H is the plume height above the surface, α is the sun angle, and d is the shadow length in the cross-plume direction. For 5 December, α is 36° and d is approximately 10 km, which yields a cloud level of approximately 7 km above the surface, or 10 km ASL. The southern portion of the plume, located approximately over the Mt. Whitney, only extends 20–50 km downstream, likely due to the time lag as the plume advects southward. A thin cloud plume is located slightly to the north, trailing from the peaks of the White Mountains and eventually merges with the main plume.

The narrow but long-extended Sierra plume is captured by a series of GOES imagery as well. Figure 2a shows a GOES water vapor image valid at 1900 UTC 30 October 2002. Warm colors indicate the presence of cloud ice or high water vapor content in the upper troposphere. A long, moist plume originates from the California coast and extends northeastward over four states: California, Nevada, Utah, and Colorado. The moist plume is more than 1500 km long and approximately 150 km wide, and the angle between the moist plume and the latitude circle is approximately 20°. Embedded in the moist plume, a much smaller plume indicated by the orange color in Fig. 2a originates from the Sierra massif and extends across the border between the states Nevada and Utah. The inner plume coincides with the cirrus plume shown in the MODIS image (Fig. 1), indicating that the stronger reflection (i.e., orange color) is likely associated with the ice particles in the cirrus plume. The animation of the GOES water vapor imagery indicates that the moisture plume originates from a moist patch located off the coast of North America (Fig. 2a), associated with strong convection over the Pacific Ocean. The moisture plume forms around 1200 UTC approximately oriented along the 40°N latitude and is much wider but shorter than that observed at 1900 UTC. It moves slowly southward while it is being stretched. The inner plume first becomes distinguishable around 1630 UTC downstream of the White Mountains (see Fig. 1 for location). As the moisture plume moves slowly southward, a more spectacular cloud plume appears downstream of the main Sierra Nevada ridge and extends farther downstream with time. At approximately 2200 UTC, the cloud plume reaches its maximum length, approximately 420 km, and extends eastward over Utah. Subsequently, the plume detaches from the Sierra Nevada ridge and then advects eastward over the Intermountain West and eventually dissipates. During the course of the plume development, as it drifts southward and extends farther downstream, the upwind edge of the cirrus plume always coincides with the main Sierra Nevada ridge prior to 2200 UTC, indicative of the importance of topographic forcing on the plume formation.

Two GOES visible snapshots are shown in Fig. 3. At 1800 UTC, the cloud plumes are evident over both the White Mountains and the main Sierra Nevada peaks. The plume located downstream of the main Sierra Nevada ridge is much brighter, and the plume trailing from the White Mountains is much longer. At 1900 UTC, the cloud plume from the main Sierra Nevada ridge becomes significantly longer than observed 1 h earlier. The southward-drifting speed and growth rate of the plume length can be estimated by comparing the two images, which are approximately 5 and 35 m s⁻¹, respectively.

b. The Southern Rockies plume

Shown in Fig. 2b is a GOES water vapor image valid at 1902 UTC 5 December 2000. Two distinct cloud patches are located immediately to the east of the Southern Rockies, separated by an approximately 100-km-wide cloud-free zone. We are particularly interested in the spectacular southern cloud plume, the left edge of which is parallel to the main massif of the Sangre de Cristo Range. The plume is approximately 500 km long and 400 km wide, and spreads over five states: Colorado, Kansas, New Mexico, Texas, and Oklahoma. Along the plume, there are two reflectivity maxima; one is located immediately downstream of the Sangre de Cristo ridge, and the other is located approximately 250 km farther downstream. The animation of the half-hour interval GOES water vapor imagery indicates that the development of the cloud plume is associated with the passing of a moist air mass advected northeastward from the state of Arizona. The cloud plume first appears in the lee of the Southern Rocky Mountain ridge around 1330 UTC, and grows rapidly in both length and width. The plume reaches its maximum in terms of the cloud cover area around 1900 UTC, and dissipates gradually afterward. The left edge of the plume coincides with the Sangre de Cristo Massif, and is fairly stationary over the whole period. The growth rate or advection speed of the plume is approximately 40 m s⁻¹ as estimated from the GOES water vapor images.

According to the GOES visible images (Fig. 4), the plume is approximately 300 km long at 1645 UTC. A well-defined foehn window is located upstream of the cloud plume, indicating strong flow descent and implying the importance of the Southern Rocky Mountain ridge in the cloud plume formation. At 1902 UTC (Fig. 4b), the plume is longer, and the small-scale variation of the brightness along the Rocky Mountain ridge is evident, likely corresponding to the variation of the ridge height. Although the two cloud plume events occur over different barriers, they share some remarkable similarities as summarized below.

• The plumes originate from high ridges, with their upwind edges coincident with high peaks throughout the two events, strongly indicating that these plumes are topographically generated.

• Both plumes are located in the upper troposphere and oriented approximately along the wind direction.

• Both plumes show multiple scale features, namely, hundreds of kilometers along the plume, and finescale variations along the ridges.

a. Sierra plume

Figure 5 shows the horizontal wind speed, potential temperature, and RH profiles derived from the 1200 UTC 30 October 2002 Oakland, California (37.75°N, 237.78°E), sounding. Based on the launch time and the GOES images (i.e., Fig. 2a), the sonde likely ascended through the moist plume shown in the water vapor images. The wind direction is northwesterly below 7 km and approximately westerly above. The tropopause is located approximately at 12.5 km, below which the troposphere shows a two-layer structure in terms of relative humidity and stability. The lower layer, from the surface to 8 km ASL, is drier and more stable with an average buoyancy frequency of 0.0126 s⁻¹. In the lower layer, the wind speed almost linearly increases from 10 m s⁻¹ at the 1-km level to approximately 20 m s⁻¹ at the 8-km level. A sharp increase of RH and decrease of stability occurs at 8 km ASL. The humidity relative to ice is close to 100% and the average (dry) buoyancy frequency is 0.009 s⁻¹ between 8 km and the tropopause (∼12.5 km). The altitude of the moist layer is consistent with the cloud-top level estimated from the cloud shadow. The air temperature in the moist layer decreases from −31°C at the bottom (i.e., 8 km) to −64°C near the top, implying that the cloud plume is primarily composed of ice particles.

Shown in Fig. 6 are the geopotential height contours, wind vectors, and RH with respect to ice (in grayscale) at 300 hPa derived from the COAMPS coarse mesh (i.e., ΔX = 27 km) valid at 1800 UTC. A pressure trough is located off the British Columbia coast, and a cutoff low is centered around the state of Montana. Between the two intense low pressure systems, there are a pair of pressure ridges: one is located over British Columbia and the other is located off the west coast of the United States. A slowly moving cold front is located between 35° and 40°N with a maximum wind speed of 30 m s⁻¹ at the 300-hPa level. The COAMPS-simulated RH field matches the GOES water vapor image reasonably well (Fig. 2a). For example, both show the moist region located off the North America coastline, and dry region located over the Southern United States. The COAMPS also captures the presence of the moisture plume, which is in reasonable agreement with the GOES water vapor images in terms of the timing, orientation, and length of the plume. Animation of the COAMPS plots (same as in Fig. 6 but for different time) indicates that the moist air is entrained into the deformation zone from the moist patch off the coast. It is stretched and extends farther inland from 1500 to 2000 UTC while it moves slowly southward, which is consistent with the series of GOES images. The wind direction in the deformation zone is westerly with wind speed between 20 and 30 m s⁻¹. It is remarkable that the COAMPS model forecasts the long, narrow moist plume correctly, considering that the plume originates from the data-sparsed Pacific Ocean. It should be noted that the inner ice cloud plume trailing from the Sierra, as shown in the GOES water vapor images, is absent from the COAMPS coarse mesh.

b. Southern Rockies plume

Vertical profiles derived from a sounding launched at 1200 UTC 5 December 2000 from Albuquerque, New Mexico (35.05°Ν, 253.39°E), located upstream of the Southern Rockies plume, are shown in Fig. 7. Similar to the Sierra plume event, the troposphere shows a two-layer structure in terms of the RH and stability. The air below 7 km ASL is much drier and more stable with an average buoyancy frequency of 0.0127 s^−1. The tropopause is located at 9.5 km, much lower than observed during the Sierra plume event. A sharp increase of the RH occurs around 7 km ASL, above which the air is more moist and less stable. In fact, the high relative humidity layer extends well into the lower stratosphere with the RH relative to ice close to 100%. Above 2 km ASL, the wind is almost strictly westerly with little directional shear and the wind speed increases from 11 m s⁻¹ at 3 km to 20 m s⁻¹ at 7 km. In the moist layer, the average wind speed is approximately 20 m s⁻¹. The air temperatures at 7 and 9.5 km are −25° and −43°C, respectively.

Shown in Fig. 8 are the geopotential height contours, relative humidity with respect to ice (grayscale), and wind vectors at 300 hPa derived from the COAMPS 27-km grid mesh valid at 1800 UTC 5 December 2000. The jet stream is located over the central United States, between a deep low pressure system over southern Canada and a pressure ridge over Mexico. Associated with the pressure ridge and a closed low located off the coast of California, a subtropical jet transports warm and moist air from the Pacific Ocean toward the southern Rocky Mountains. The wind direction over the states of New Mexico and Arizona is southwesterly and turns into westerly in the vicinity of the Southern Rocky Mountain ridge. A second moist air mass over the North America is located farther north associated with a shortwave embedded in the Canadian trough. A dry air zone is located in the deformation zone, which separates the two moist air masses. The cloud patches in Figs. 2b and 4 and the moist air masses in Fig. 8 match each other reasonably well.

a. Cross-sectional analysis

Shown in Fig. 9 are the vertically integrated cloud ice [i.e., VIC(x, y) = ∫ ρ_aq_i dz] and horizontal wind vectors at 9 km ASL derived from the innermost mesh valid at 1900 UTC 30 October 2002. Areas with model terrain above 3 km are hatched. A well-defined plume trailing from the main Sierra ridge is evident. The plume is more than 250 km long and 70 km wide, and there is a small angle (∼20°) between the plume orientation and the wind direction at 300 hPa (i.e., westerly). The estimated plume growth and southward-drifting speeds are 25 and 6 m s⁻¹, respectively, which are approximately in agreement with the estimation from the series of GOES images. The cloud ice mixing ratio, ice liquid water equivalent potential temperature, wind speed, and relative humidity (relative to ice) in a vertical cross section oriented along the plume (see Fig. 9 for location) are shown in Fig. 10a. The ice liquid water equivalent potential temperature is defined as (Tripoli and Cotton 1981)

View Expanded

where C_p is the air specific heat at a constant pressure, L_iυ = L_lυ + L_f , L_lυ is the latent heat of vaporization, L_f is the latent heat of fusion, and q_l and q_i are mixing ratios of liquid water and cloud ice, respectively. Along this section, the main ridge width is of the order of 100–150 km and the lee slope is fairly steep. A cloud ice plume is located between 8 and 11 km, which is consistent with the cloud level estimated from the cloud shadow. While the cloud ice is present both upstream and downstream of the ridge, the terrain enhancement effect is still evident. The equivalent potential temperature field indicates the presence of windward blocking as indicated by the interception of isentropes with terrain. The RH field indicates that a moist layer is located between 8.5 and 11 km, where the air is ice saturated or nearly saturated even on the upstream side of the Sierra Nevada ridge. An inertia–gravity wave (IGW) is launched from the highest peak and penetrates the moist layer in the upper troposphere. Near the cloud plume level, the IGW is characterized by a sudden updraft over the peak and a slow descent in the tail. The ice saturation is a sufficient condition for crystal formation in a cold environment in the model (Fletcher 1962), which is a necessary but not a sufficient condition in nature according to observational studies (e.g., Heymsfield and and Miloshevich 1995). This issue will be further discussed in the coming sections. Nevertheless, the relative humidity seems to be increased in the lee associated with IGW-induced updrafts.

In Fig. 11, a widespread cloud ice plume originating from the Sangre de Cristo ridge is evident. The plume extends more than 300 km westward and is approximately oriented along the wind direction. The COAMPS-forecasted cloud ice field agrees with the GOES imagery very well in terms of the location, orientation, and size of the cloud plume. Figure 12 shows the vertical cross section of the equivalent potential temperature, cloud ice, and wind speed, oriented along the plume. Areas with the relative humidity relative to ice greater than 90% are hatched. The highest peak of the Sangre de Cristo ridge is about 3 km ASL. Again, there is a near-saturated layer located approximately between 8 and 12 km. As airflow passes the ridge, IGWs are launched, the amplitude of which decays with the increasing altitude over the ridge. To the right (i.e., downstream side) of the ridge, the cloud ice mixing ratio is significantly larger, indicative of the crucial role of the Southern Rocky Mountains in the plume formation.

b. Trajectory analysis

Figures 13 and 14 show some results from trajectory analysis using the 3-km grid data. The trajectory of a Lagrangian parcel is computed using x_i+1 = x_i + v(x_i, t_i)Δt, where x = (x, y, z) is the position vector, v = (u, υ, w; t) is the four-dimensional wind field linearly interpolated from half-hour interval COAMPS 3-km grid data, and Δt is the time interval.

For the Sierra plume event, parcels are launched at 1900 UTC from 36.55°Ν, 243.5°E, which is located within the cloud plume and approximately 150 km downstream of the main Sierra ridge. The initial parcel altitudes are 6, 7, 8, 9, and 10 km, respectively, with Δt = −0.25 min; a negative time interval corresponds to a reverse trajectory. The parcel altitudes, relative humidity (with respect to ice), and cloud ice mixing ratio along the trajectories are plotted versus horizontal distance. The terrain higher than 2 km ASL and underneath the 6-km parcel is plotted for reference. Clearly, for all the parcels, strong ascent (∼600–1200 m) occurs over the lee slope of the main Sierra Nevada ridge and the parcel relative humidity increases accordingly. The parcels initially from the 7-, 8-, and 10-km levels are drier (RH < 65%) upstream of the Sierra Nevada ridge and become ice saturated over the highest peak. The parcel originating upstream from the 9-km level has greater RH and becomes saturated over the windward slope associated with weak upslope ascent. The parcel experiences a sharp descent over the highest peak where the RH decreases to 80%, and becomes saturated subsequently over the lee slope associated with a sudden ascent.

Theoretically, in stratified flow, a parcel that temporarily departs vertically from its equilibrium level tends to return to its level of origin because of buoyancy force. However, the parcels launched from the 7–10-km levels stay aloft over more than 150 km downstream, and, correspondingly, remain saturated or near saturated. The parcel launched from the 6-km level (i.e., below the cloud plume) is not saturated and experiences a net descent downstream. The topographic control of the cloud ice formation is evident in Fig. 13c. For parcels initially launched from 7 to 10 km (i.e., within the cloud plume), the highest peak of the Sierra ridge separates the no-ice upstream state to ice-contained state downstream. The ice mixing ratio of the parcel from 8 km indicates an oscillation with a horizontal wavelength of approximately 50 km.

A similar set of trajectories computed from the Southern Rockies plume simulation are shown in Fig. 14. The parcels are launched at 1900 UTC from 35.5°N, 258°E, which is located within the cloud plume and approximately 300 km downstream of the Rockies, and traced back to upstream using the backward trajectory calculation method with Δt = −0.25 min. Clearly, the parcels launched from 7, 8, and 10 km experience ascent over the lee slope of the Southern Rocky Mountain ridge and stay aloft for more than 300 km downstream. The parcels from the 8–10-km layer are ice saturated downstream of the Rocky Mountain ridge where cloud ice appears accordingly. In addition to the net ascent, some parcels oscillate in the vertical as they drift downstream, with a horizontal wavelength of approximately 100 km. Accordingly, the cloud ice mixing ratios exhibit similar oscillation for parcels from the 7–10-km layer.

In summary, the trajectory analysis clearly indicates the topographic control of the cloud ice formation during the two plume events examined. Cloud ice is generated in the lee of the major ridges associated with a sudden ascent during which parcels become ice saturated. Farther downstream, parcels may stay aloft with a net positive vertical displacement or oscillate in the vertical, likely associated with inertia–gravity waves as will be discussed in the next section.

c. Microphysical aspects

The domain-averaged crystal number concentrations, crystal diameters, and terminal velocities for the two control simulations are listed in Table 1 and the corresponding frequency distributions of the grid-mean crystal diameters and terminal velocities are shown in Fig. 15. The domain-averaged quantities are derived from the COAMPS simulations and averaged over grid points with q_i > 5 × 10⁻⁶ kg kg⁻¹, approximately corresponding to crystals with a grid-mean diameter of less than 10 μm. It is interesting that the domain-averaged crystal properties and the frequency distributions of grid-mean diameters and vertical velocities derived from the two control simulations are almost identical. For approximately 80% of the grid points, the grid-mean crystal diameters are between 50 and 150 μm and the terminal velocities are between 2 and 10 cm s⁻¹. As expected for high cirrus clouds, the crystal sizes are relatively small for both plumes and correspondingly the vertical velocities are small as well, which is consistent with the marked length of the two plumes. Despite the simplicity of the ice-phase physics parameterization used in COAMPS, the simulated crystal number concentrations, sizes, and vertical velocities are comparable to those aircraft measurements of cirrus clouds at a similar altitude (e.g., Heymsfield 1975; Heymsfield and Iaquinta 2000).

To examine the sensitivity of the cirrus plumes to microphysical parameterization, additional simulations of the Southern Rockies plume event have been carried out with the numerical setup identical to the control run except for the coefficient n_o in Eq. (1) and k_t in Eq. (3). Results from two pairs of such simulations with k_t = 3040 and 30.4, and n_o = 1 and 10⁻⁴ m⁻³, respectively, are included in Table 1 and Fig. 16. Because the crystal number concentration is only a function of the air temperature, the domain-averaged number concentration is almost independent of k_t (Table 1). Corresponding to the increase of k_t by a factor of 10, the domain-averaged terminal velocity is increased approximately by a factor of 8, and the mixing ratio of cloud ice and diameter are significantly reduced, likely due to the fallout of large crystals. Compared to the control run, the cloud cover area as indicated by the vertically integrated ice shrinks considerably (Fig. 16a). As k_t is decreased to one-tenth of its original value, the domain-averaged mixing ratio and crystal diameter increase slightly and the terminal velocity decreases to approximately one-tenth of that of the control run. As expected, associated with much slower falling speed, the cloud cover area is larger, and the decay of denser clouds is much slower (Fig. 16b). For a smaller n_o, the crystal number concentration is proportionally smaller, and the domain-averaged crystal diameter and terminal velocity increase approximately as n^−1/2_o. Correspondingly, the plume is smaller than that in the control run (Fig. 16c). For a larger n_o, the crystal diameter and terminal velocity decrease approximately as n^−1/2_o due to the increase of crystal number concentration. Associated with slower fallout of crystals, the simulated plume is larger and more intense (Fig. 16d) similar to the simulation with k_t = 30.4.

a. Inertia–gravity wave dynamics

For stratified unidirectional flow with a constant wind speed U, the linear steady IGW equation can be written as (Queney 1948)

View Expanded

where w is the vertical velocity and f is the Coriolis parameter. Equation (5) has been solved for stratified airflow past an idealized ridge using the Fast Fourier Transformation (FFT) method (Smith 1980). The computational domain is two-dimensional including 1024 points in the horizontal and the resolution is 1 km with periodic boundary conditions applied along the lateral boundaries. At the surface (i.e., z = 0), we have w = Uh_x, where h(x) = h_mexp(−x²/a²) is a Gaussian ridge defined by a ridge height h_m and a half-ridge width a. The atmosphere is separated into three discrete layers with a constant buoyancy frequency N and ambient wind speed U in each layer. The lowest two layers are used to represent the troposphere and the third stable layer is assumed to be infinitely deep with a radiation condition applied at the top. The layer depths and other parameters, derived from the two soundings shown in Figs. 5 and 7 are listed in Table 2 for reference. The ambient flow is in geostrophic balance (see Smith et al. 2002 for more details about the three-layer linear wave model). To illustrate the dynamical role of IGWs in the cloud plume formation, four linear IGW solutions associated with stratified flow past an idealized ridge are shown in Fig. 17. Figure 17a shows a solution with f = 10⁻⁴ s⁻¹ and an asymmetric ridge with h_m = 1000 m, and a = 60 km in the windward side and 15 km in the lee side. The asymmetric ridge is used to mimic the Sierra ridge cross section shown in Fig. 10. Between 7 and 10 km, the vertical displacement in the lee of the ridge peak indicates a sharp ascent and then a slow descent downstream. The length of the positive vertical displacement area (PVDA; shaded) is approximately 120 km. Clearly, the level to have relatively large PVDA is located at 1.5λ_z, where λ_z = 2πU/N is the vertical wavelength of hydrostatic waves.

The presence of the PVDA in the lee of the ridge is consistent with the real-data COAMPS simulations. Especially, the sharp-ascent and slow-descent pattern might be critical for the formation of the long cloud plumes. The second wave crest is located approximately 400 km downstream of the ridge with its amplitude reduced to one-fifth of the first wave. In the absence of rotation, the dominant waves are hydrostatic and characterized by a slow descent and sharp ascent pattern between 7 and 10 km ASL in the vicinity of the ridge and the positive vertical displacement area in the rotational solution is absent (Fig. 17b). The positive vertical displacement area is still present in the solution with a less steep lee slope (i.e., a = 60 km on both upstream and lee sides, Fig. 17c) and the leading edge of the PVDA is gentler accordingly, indicative of weaker ascending motion. Figure 17d shows a solution with the Albuquerque sounding and a narrower ridge (i.e., a = 30 km). Again, the solution indicates the presence of a PVDA with a sharp ascent and subsequent slow descent. The along-wind dimension of the PVDA is approximately 200 km.

There are two velocities relevant to the development of the plumes: the advection speed U and the horizontal component of the IGW group velocity C_gx. In two-dimensional (i.e., x–z) hydrostatic limit, the dispersion relation of IGWs can be written as

View Expanded

where m, k are vertical and horizontal wavenumbers, and the ± signs correspond to downstream- and upstream-propagating waves, respectively. We are only interested in waves that are stationary relative to the terrain (i.e., the horizontal wave phase speed C_px = ω/k is zero), which gives

View Expanded

Using (6), (7), and C_gx = ∂ω/∂k, we obtain

View Expanded

where R_k = Uk/f is the Rossby number based on the horizontal IGW wavenumber. If we define the Rossby number as R_L = U/( fL), where L is the wavelength, the nondimensional horizontal component of the group velocity is (4π²R²_L)⁻¹ ≃ 0.025R⁻²_L. As an example, if we choose U = 20 m s⁻², L = 350 km (estimated from Fig. 14a), and f = 10⁻⁴ s⁻¹, it follows that R_L ≃ 0.6 and C_gx ≃ 1.4 m s⁻¹. Clearly, the observed rates at which the cirrus plumes extend downstream are comparable to the cloud-level advective speeds, and much faster than the estimated IGW horizontal group velocity.

b. Microphysics

Instead of examining the microphysical process in detail, we try to put this study in the proper perspective by comparing with relevant published results. Previous studies indicated that the nucleation processes in the cold environment are sensitive to both air temperature and relative humidity (Heymsfield and Miloshevich 1995). In the upper troposphere, heterogeneous nucleation was found relatively inefficient (Hobbs and Rangno 1985; Heymsfield et al. 1991). For example, when the air temperature is lower than −36°C, liquid droplets were absent in aircraft in situ measurements, suggestive of the domination of homogeneous ice nucleation in high cirrus clouds (Sassen and Dodd 1988; Heymsfield and Sabin 1989). According to the aircraft measurements made in cirrus during the First International Satellite Cloud Climatology Project (ISCCP) Research Experiment (FIRE) II, the lower bound on the relative humidity required for cirrus formation RH_c ≈ −RH_hn − 10 (Heymsfield and Miloshevich 1995). Here RH_hn is the peak relative humidity with respect to water in wave clouds, which decreases from 100% for temperature above −39°C to 73% at −56°C approximately following

View Expanded

where T is the temperature in degrees Celsius. Here we consider an air parcel that ascends adiabatically. The minimum vertical displacements δ_c and δ_hn required for the parcel to reach RH_c(T) and RH_hn(T) can be obtained using (9). The derived δ_c and δ_hn and the corresponding RH with respect to water RH_w and ice RH_i using the Oakland and Albuquerque soundings are shown as a function of the initial parcel altitude (Fig. 18). Clearly, for the Oakland sounding, the minimum vertical displacement required to form cirrus (i.e., the δ_c) is more than 1 km for air parcels initially located below 8 km ASL, and is much smaller (i.e., ∼300 m) in the moist layer. Especially near the tropopause, δ_c is close to zero, which is consistent with the presence of thin cirrus clouds upstream of the Sierra ridge (Fig. 1). The corresponding saturation ratio with respect to ice is between 110% and 130%. The minimum vertical displacement δ_c computed using the Albuquerque sounding is of the order of 700 at the bottom of the moist layer (i.e., ∼7 km) and linearly decreases to zero near the tropopause (i.e., ∼9.5 km). The corresponding relative humidity is in the range of 120%–140% in the moist layer.

Karcher and Strom (2003) found that vertical air motion is a key factor in determining ice crystal concentrations and size distribution in cirrus clouds. Associated with strong updrafts (i.e., w > 0.1 m s⁻¹), most young cirrus clouds typically contain a high number density of small (diameter <20 μm) ice crystals, and likewise, with slower updrafts, cirrus clouds more likely contain low concentrations of larger crystals. The presence of large numbers of small ice crystals in mountain wave–induced cirrus clouds associated with rapid cooling have been reported by other observational and numerical studies as well (e.g., Lin et al. 1998). It was demonstrated by Heymsfield and Miloshevich (1993) that corresponding to the increase of vertical motion from 0.25 to 2 m s⁻¹, the ice crystal number concentrations approximately increase by 100 times. Using the vertical displacement in Fig. 18 and the 2D linear steady relation w_c = U∂δ_c/∂x = 2πUδ_c/L, where L is the horizontal wavelength corresponding to the vertical motion, we can estimate the minimum updraft (i.e., w_c) for homogeneous nucleation and cirrus formation. When U = 25 m s⁻¹, L = 60 km, and δ_c = 300 m (i.e., approximately corresponding to the Sierra plume event), w_c is approximately 0.8 m s⁻¹. Similarly, using U = 30 m s⁻¹, L = 30 km, and δ_c = 300 m (relevant to the Southern Rockies plume), we obtain w_c ∼ 2 m s⁻¹. The simulated updraft maxima over the high peaks and at cloud levels are larger than the estimated w_c values for the two plume events. According to linear mountain wave theory, for given wind and stratification profiles, the mountain wave amplitude (i.e., vertical velocity w or vertical displacement δ) is proportional to the terrain height and the threshold values of δ_c and w_c estimated here correspond to certain critical mountain heights to generate these perturbations, implying that these cloud plumes only form over relatively high ridges.

Apparently, the terminal velocities of cirrus crystals are of primary importance in determining the decay of the cirrus plumes. Once falling out of clouds, ice particles are subjected to sublimation in an ice-subsaturated environment. While laboratory and observational studies have highlighted the dependence of crystal terminal velocity on the crystal shapes, in general, the terminal velocity is approximately proportional to the crystal diameter (e.g., Fig. 3 of Heymsfield and Iaquinta 2000). As revealed by the MODIS visible image, the length of denser clouds, likely associated with larger crystals, is ∼50 km and the length of the whole cloud plume is ∼400 km. Using U = 25 m s⁻¹, we obtain two time scales: 2000 and 16 000 s. Assuming that the cloud depth is 2 km, the average fallout velocity for large crystals is approximately 1 m s⁻¹, and for small crystals, the average fallout velocity is approximately 0.12 m s⁻¹, corresponding to crystal sizes of 200–400 μm and <50 μm, respectively. The survival of crystals falling into subsaturated environment has been studied by Hall and Pruppacher (1976). They found that the survival time of crystals in an ice-subsaturated environment is very sensitive to the relative humidity; the higher the RH, the longer the crystal can survive. In addition to fallout, crystals in cirrus plumes could also sublimate in an ice-subsaturated environment created by wave-induced descent. The relatively deep moist layer (∼2 km) observed during the two plume events and the slow descent associated with IGWs apparently help maintain the high RH environment around small crystals and likely contribute to the maintenance of the long plume tails.

c. Thermodynamics and parcel argument

While the presence of the long PVDA in the IGW solutions is likely relevant to the cloud plume formation, the application of linear wave theory to the cirrus problem could be complicated by diabatic cooling or warming associated with ice physics. It has been shown in previous studies that dry wave theories are still applicable to a moist atmosphere involving diabatic processes if the dry buoyancy frequency N is replaced with a moist buoyancy frequency N_m (Lalas and Einaudi 1974). In the presence of cloud ice, we can define a moist N_m as N ²_m = gd(lnθ_i)/dz. This approach may become problematic due to the fallout of ice crystals or microphysical hysteresis. The term hysteresis here is used loosely to refer to the time scale difference in ice crystal growth and sublimation (i.e., sublimation is slower), or the relative humidity difference required for crystal formation (RH_i > 110% in the upper troposphere according to Fig. 18) and for sublimation (RH_i < 100%). The use of moist N is also inappropriate as the microphysical-related time scale is comparable or larger than N⁻¹.

Following a Lagrangian parcel, the increase of parcel potential temperature due to the growth of ice crystals at the cost of water vapor is

View Expanded

where θ is the potential temperature, T is the parcel temperature. For dq_i = 0.15g kg⁻¹ (estimated from Figs. 12 and 13), θ = 320 K, the corresponding potential temperature increase is ∼1 K, which is fairly significant considering the relatively weak stability in the moist layer. For example, if the potential temperature gradient in the ambient flow is 3 K km⁻¹ (i.e., N ∼ 0.01 s⁻¹), a 1-K increase in potential temperature allows the parcel to reach a new equilibrium level, which is approximately 300 m above its original level.

To further illustrate the effect of ice cloud hysteresis on vertical motion, we consider a moist Lagrangian parcel, the vertical motion of which is governed by

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where θ_a and θ_p are the ambient and parcel potential temperatures, respectively, and w is the vertical velocity. Initially, the air parcel is in equilibrium (i.e., θ_p = θ_a = θ_o), and the vertical velocity is w_o > 0, corresponding to mountain-induced ascent. A few possible solutions to Eq. (11) are schematically shown in Fig. 19 and elaborated below. It is assumed that instant conversion from water to ice occurs whenever the relative humidity relative to water reaches a certain critical value of RH_c.

1. When the initial updraft is weak, the parcel maximum vertical displacement is less than δ_c, the minimum vertical displacement required for the parcel to reach RH_c(T), and no cirrus cloud is generated.

2. If w_o is sufficiently large and the parcel is lifted higher than the threshold δ_c, ice forms and the air parcel potential temperature increases (i.e., path A–B–C). The parcel potential temperature keeps increasing associated with further ascent and more ice conversion from water vapor, until w is reduced to zero. If ice particles are large enough and fall out of the parcel, the parcel will maintain its potential temperature θ_p(C) during its descent phase until a new equilibrium is reached with θ_p(D) = θ_p(C) > θ_po. Associated with a net increase in potential temperature, the new equilibrium level is higher than the initial level, implying a permanent increase in altitude. The cloud plume length is equal to U(τ_AC + τ_f), where τ_AC is the time for the parcel to travel from A to C and τ_f is the ice particle fallout time. As τ_AC is usually small, and so is τ_f for large particles, a short cloud plume is generated in this case.

3. Same as 2 with the exception that the ice particles are too small to fall out. The parcel follows path A–B–C–D and reaches a temporary equilibrium at point D. Then associated with sublimation due to IGW-induced descent, θ_p decreases and the parcel gradually descends back to its original level (i.e., A). The cloud plume length is approximately U[τ_AD + max(τ_sub, τ_d)], where τ_AD is the time for the parcel to travel from level A to D, and max(τ_sub, τ_d) is the larger one of the sublimation time scale τ_sub and the IGW-induced descent time scale τ_d. In this case, a long plume composed of small-sized particles is generated.

4. Initially, the parcel is saturated relative to ice and its RH relative to water is still too low for homogeneous nucleation (i.e., RH < RH_c). As the parcel is lifted above the threshold level, similar to the scenario 3, the parcel travels through states A′–B′–C′–D′. Subsequently, slow sublimation and descent occur until a new equilibrium state E′ is reached. The net increase of altitude from A′ to E′ is due to the supersaturation of the parcel relative to ice at the initial point A′, and the cloud length in this case is infinity.

According to the satellite images and model trajectory analysis, processes 2, 3, and 4 are likely relevant to the generation and maintenance of the studied cirrus plumes.