3.1. The Jumping Cirrus Phenomenon in the Simulated Storm
 Before the model results are discussed, it is worthwhile to make clear that the appearance of the jumping cirrus phenomenon in the simulation does not imply that it actually occurred in the CCOPE storm. It merely indicates that the jumping cirrus phenomenon could occur in a severe storm such as this one, and if it does it is most likely due to the mechanism suggested here. This is because that the model solves mathematical equations based on well-known physics and these physics can explain various phenomena seen in the simulation results. Thus the purpose of this study is to point out that the jumping cirrus phenomenon as described by Fujita could happen under favorable environmental conditions and can be explained by simple dynamics.
 Figure 1 shows a series of 12 snapshots of the RHi (relative humidity with respect to ice) profiles in the central east-west vertical cross-section (y = 27 km) of the simulated storm every 120 s from t = 1320 to 2640 s. High RHi regions represent locations of high probability of ice crystal formation and hence is a reasonable approximation of the cloud boundary, especially the cloud top region [Wang, 2003]. To focus on the cloud top region, these snapshots are windowed to 10–20 km vertically and 20–55 km horizontally, with the vertical scale stretched in these views. The range of the vertical axis is from 10 to 20 km and that the general shear direction is from left to right (west to east).
Figure 1. Snapshots of the RHi profiles in the central east-west cross-section of the simulated storm from t = 1320 s to 2640 s.
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 At t = 1320 s, the storm top exhibits a two-wave pattern: one crest located at the main updraft region (x ∼ 30 km) and the other at x ∼ 40 km. At this stage the overshooting is not yet well developed and the highest point of the cloud is only slightly higher than the tropopause at 12.5 km [Johnson et al., 1993]. However, the wavy nature of the storm top is already obvious. At t = 1440 s, a cloudy patch starts to emanate from the bulge in the cloud top below. This patch is the precursor that eventually develops into full-fledged jumping cirrus. The white arrow pointing at x ∼ 34 km indicates the approximate position of the left (west) edge of the patch. At the same time, the overshooting top subsides, changing from a height of ∼13 km to ∼12.5 km, a drop of ∼500 m. This seems to correspond to what Fujita  described as the “collapse of the overshooting dome”. While the overshooting top is subsiding, the wave crest located at x ∼ 40 km starts to bulge up and tilt upstream. At 1560 s, a “jumping cirrus” in the form of a cirrus tongue has developed with its front edge located at x ∼ 32 km and reaching an altitude of ∼15 km. The cirrus tongue is already located higher than the overshooting top and is moving upstream. Note also that a third wave crest appears at x ∼ 48 km at this time. Thus the average “wavelength” of the waves on cloud top is approximately 9 km, although the distance between the first two upstream wave crests is only 6–7 km. The “tail” end of the jumping cirrus seems to originate from the detachment from the third wave crest.
 As time goes on, the cirrus reaches further west and higher altitude as can be seen by the locations of the white arrows at the front edge. Since the altitudes of the jumping cirrus are both ∼15 km at 1560 and 1680 s, the maximum altitude probably occurred somewhere in between these two times. This upstream and upward motion corresponds to what Fujita described as the “cirrus cloud which jumps upward from behind the overshooting dome”. This ascending sequence of the jumping cirrus lasts about 6 min within which the cirrus rises from z ∼ 12 km to ∼15 km. The average vertical speed of the jump is therefore about 8 m s−1. Considering that this altitude is well within the lower stratosphere where normal vertical motion is very weak, this is a substantial vertical speed and certainly justified to be described as “jumping”. The development of the simulated cloud top up to this stage seems to verify Fujita's description of jumping cirrus.
 Fujita did not give descriptions of what happened to the cirrus after jumping upward and upstream. The model results provide additional information for possible development in the later stage. After reaching its maximum height at t ∼ 1680 s, the tongue subsides gradually but still extends to further upstream, reaching x ∼ 30 km and 29.5 km at t = 1800 s and 1920 s, respectively. Since the cirrus has moved a horizontal distance upstream of about 5 km from t = 1440 s to 1920 s, the average horizontal speed is therefore about 10 m s−1, comparable to the vertical speed. This by no means says that the speed is uniform; rather the speed is greater initially and then decreases.
 Afterwards, the cirrus becomes thinner to resemble a plume and the left half of it is almost detached from storm anvil below. The cirrus plume eventually becomes unstable and breaks into two parts. The western part seems to collapse on, and merge with, the overshooting dome. This could correspond to what Fujita  described about the stratospheric cirrus veiling over the overshooting dome.
 The eastern part of the cirrus, which is attached with the anvil up to 2160 s, becomes gradually lifted and detached at 2280 s, orientating itself nearly parallel to the anvil. The detached cirrus plume thins and drifts downstream. It becomes nearly invisible after 3600 s. As the cirrus plume dissipates, the overshooting dome becomes more prominent, as can be seen from the development in the period 2040–2640 s. The jumping cirrus phenomenon only occurred once in the entire 150-minute simulation.
3.2. The Mechanism for Jumping Cirrus Formation
 What is the mechanism responsible for the formation of jumping cirrus as described in the preceding subsection? First of all, the apparent jumping motion towards upstream is only true in the relative sense. The frames in Figure 1 are plotted relative to the storm. This is because in the simulation the storm is moving to the east at a speed of about 30 m s−1. We need to subtract this mean motion from the computed winds in order to keep the storm core remaining more or less in the center of the computational domain. So the apparent “upstream” motion of the cirrus is only true relative to the storm. Fujita  reported the cirrus' motion relative to the overshooting dome, which is also a storm-relative description. In view of the 30 m s−1 mean wind subtracted from the computed winds, the horizontal motion of the jumping cirrus simulated in the present study is moving to the east at ∼20 m s−1 relative to the earth surface.
 On the other hand, the vertical speed of the jumping cirrus is unaffected by the above adjustment. Thus the 8 m s−1 upward speed of the cirrus mentioned in the preceding subsection is the true speed.
 Careful analysis of the model results shows that the jumping cirrus forms as a result of cloud top gravity wave breaking. It is well known that severe storms excite gravity waves [e.g., Alexander et al., 1995; Lane et al., 2001, 2003]. Under sufficient unstable conditions, wave breaking can occur that may result in part of the storm, especially the cloud top, becoming detached and ejected upward into the lower stratosphere. The same wave breaking mechanism is also responsible for the formation of the jumping cirrus here. This can be seen from Figure 2 where the central cross-section of water vapor mixing ratio (qv) at 1680 s overlaid with potential temperature (θ) contours are plotted. The 380 K θ-contour shows clear sign of wave breaking. Similar plots with additional overlay of wind vectors in the wave-breaking regions are shown by Wang .
Figure 2. Snapshot of the qv profiles with overlaid potential temperature (θ) contours in the central east-west cross-section of the simulated storm at t = 1680 s.
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 Fujita's [1982, 1989] observation of the sequence that the jumping cirrus occurred after the overshooting dome collapsed is also reproduced by the simulation. This seems to indicate that the wave energy associated with the dome collapsing propagates downwind and contributes to the breaking. At present we don't know whether this is the case, and if so, the magnitude of the wave energy necessary to cause the breaking. If this is indeed the case, it would suggest that if the wave energy is dissipated more efficiently in the overshooting area, presumably due to a combination of the conditions inside the storm and the stability above the dome, wave breaking and hence the jumping cirrus downwind would not occur. This agrees with the later development of the simulated storm which shows that the overshooting dome rose higher and wave breaking even occurred on top of the dome, but no more jumping cirrus as defined here occurred.
 Lane et al.  used a very high-resolution, two-dimensional model to perform a simulation of a thunderstorm using the sounding in Bismarck, North Dakota in 1997. They found that the upstream-propagating gravity waves break. Their analysis indicates that the wave breaking is due to the build up of a local critical layer in the cloud. This is also the probable cause of wave breaking in the simulated CCOPE storm reported here. The wave analysis of the present case is being conducted and the results will be reported in the near future.
 Since the jumping cirrus occurs in regions of high instability, the turbulence level in the cloud is also high and there will undoubtedly be mixing with the stratospheric air. The sighting of jumping cirrus thus indicates the tropospheric air and moisture being injected into and mix with the stratospheric air, a troposphere-to-stratosphere transport process. It is clearly diabatic, as the potential temperature is not conserved during the transport process. A more thorough discussion of this subject is given by Wang .
 The three different jumping cirrus categories by Fujita  appear to be the same phenomenon occurring either at different intensity scale or in a slightly different cloud top environment (such as stratification in the stratosphere, the wind shear, etc.) Other than that, the same wave breaking mechanism seems to explain the main characteristics of all three. Among the three categories, the ‘geyser’ cirrus appears to be the most vigorous variety, as it can go up 3–4 km above the anvil. Although Fujita did not associate the geyser cirrus with upwind motion, it is clear that the geyser column tilts upstream from the photograph he provided [Fujita, 1989, Figure 28].
 Setvák et al.  reported the sighting of a smaller scale jumping cirrus on 24 May 1996 late afternoon from an airplane above Alabama and Georgia. Figure 3 shows a side-by-side comparison between the photograph of the jumping cirrus taken by Setvák et al.  and the rendered RHi 30% contour surface of the simulated storm top at 1440 s. The bulge to the west of the overshooting top in the simulated cloud top strikingly resembles the photographed jumping cirrus in the relative location, the upstream-leaning orientation and the surge-shape. This resemblance lends more weights to the theory of jumping cirrus as described above.
Figure 3. (Left) Jumping cirrus photographed by Martin Setvák on 24 May 1996 late afternoon from an airplane above Alabama and Georgia (Courtesy of Martin Setvák). (Right) RHi 30% contour surface of the simulated storm at t = 1440 s. The vertical dimension is enhanced to match the perspective view of the photograph.
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