Observation of Falling Motion of Columnar Snow Crystals

Transcription

1 276 Journal of the Meteorological Society of Japan Vol. 54, No. 5 Observation of Falling Motion of Columnar Snow Crystals By Masahiro Kajikawa Department of Earth Science, Akita University, Akita, Japan (Manuscript received 21 April 1976, in revised form 28 June 1976) Abstract Free-fall pattern and velocity of columnar snow crystals were observed by the stereophotogrammetric method. The experimental formula between falling velocity and mass was obtained for the various types of columnar crystals. The rotation about vertical axis and the oscillation in vertical plane were observed for all of the rimed columnar crystals. The period of oscillation was about one-half that of rotation and the non-dimensional frequency of rotation increased in proportion to Reynolds number. Although the rimed columns fell with vertical path, about 50% of the rimed needles and sheaths fell with clear spiral trajectory. 1. Introduction For the basic research of formation of snowflakes and riming phenomenon of snow crystals, it is considered that the observation of threedimensional motion of crystals in the atmosphere comes up as an important problem. There are several measurements of the falling velocity (the vertical component of falling motion) of snow crystals (for example, Nakaya and Terada, 1935; Zikmunda and Vali, 1972; Kajikawa, 1972; Locatelli and Hobbs, 1974; Kajikawa, 1975). However, the available data of velocity for columnar crystals are not enough for many purposes. Moreover, there are few observations of the characteristic feature of falling motion, although the laboratory model experiments in relation to the motion of snow crystals were performed by many researchers (for example, Jayaweera and Mason, 1965; Podzimek, 1965; Jayaweera and Mason, 1966; Podzimek, 1968). Sasyo (1971) observed the motion of individual snow particles falling through still air using a stereophotogrammetric method and investigated statistically the feature of motions. But, in his observation the identifying a snow particle to its corresponding trajectory on photographs was not performed. Zikmunda and Vali (1972) observed the failing motion of rimed columnar crystals using a stereoscopic camera system. They mentioned that the mass distribution and flow disturbances by surface features are to be responsible for the complicated fall patterns involving rotation and oscillation, however the data are insufficient to consider this type of motion generally. In the present study, the three-dimensional motion and shape of individual columnar snow crystals were observed simultaneously using a stereoscopic camera system with stroboscopic illumination, and the characteristics of motion were discussed with the several parameters of crystals. The observation was carried out at the Mt. Teine observatory of Hokkaido University, the altitude being 1024 m, during the winter of Method of observation The apparatus and stereoscopic camera system used in this observation are shown in Fig. 1 and they were placed in a cold observation room. This apparatus is made of two parts, a duct (A) for the fall of snow crystals and a wooden box (B) for the observation of falling motion. To keep the air in it stable, this observation box was made airtight and its bottom was slightly cooled by freezing mixture. As a result of this regard, the air temperature of lower part of this box was always about 1.5* lower than that of observation room. One by one, a suitable columnar snow crystal was picked up with a piece of wood from many snow crystals received on a black cloth and it

2 October 1976 M. Kajikawa 277 Fig. 1 Apparatus, stereoscopic camera system and coordinate axes. was dropped into the duct. For the measurement of three-dimensional motion, the falling crystal illuminated by stroboscopic light was photographed by means of a stereoscopic camera system with the base line of 16 cm. Examples of the pair of photographs are shown in Fig. 2 (Case A) and Fig. 3 (Case B). The vertical fall distance from the inlet of duct to observing area was about 1 m, so that the natural condition of free-fall motion of crystals was attained before coming into the field of vision (16.8 *9.4 cm) of camera system. The fallen crystal was caught on a sampling glass plate which was covered with white vaseline, and it was microphotographed for determing the type and size. Then the crystal was melted by a feeble electric heater placed under the microscopic stage and obtained droplets were microphotographed again. The mass of crystal was calculated from the volume of those hemispherical droplets (see Fig. 2). Stereophotographs were analysed by the same method as Sasyo (1971) and also observed by a Fig. 2 A example (Case A) of stereoscopic photographs (above) which were taken with the time interval of 1 / 100 sec. Fallen rimed needle, before melting (below left) and after melting (below right).

3 278 Journal of the Meteorological Society of Japan Vol. 54, No. 5 Fig. 3 Stereoscopic photographs of Case B, taken with the time interval of 1/100 sec. Fig. 4 Measured factors of falling columnar snow crystal. O'X' is parallel to OX of Fig. 1. Fig. 5 Measured values by the stereophotogrammetric method of the length along c-axis (l=1.96 mm, in microphotograph) of a rimed needle. stereoscope. Because the optical axes of cameras put on horizontal level were parallel, the normal photogrammetry equations were available in the analysis of three-dimensional motion. The optical center of camera 1 is chosen as the origin (0) of a rectangular coordinate system (X, Y and Z) in space. The coordinates of both ends (C1 and C2 in Fig. 4) of columnar crystals were calculated from the corresponding points on pair of photographs enlarged to cabinet size. The direction angle (~) of c-axis in the horizontal projection, the deviation angle (e) of c-axis from the horizontal plane and the coordinates of center (C) of crystals were calculated by using the coordinates of C1 and C2 for successive points, in every 1 / 100 sec intervals. Although the error of this stereophotogrammetric measurement is composed of many factors, it is very difficult to estimate all of these errors exactly. However, the accuracy of experimental results is able to examine by the comparison of following two values of length (l), because l can be measured from both of the photogrammetric analysis and the microphotograph, independently. For example, l of rimed needle (1.96 mm in microphotograph) was measured by the photogrammetric analysis of measuring points of 32 on the pair of photographs, as shown in Fig. 5. Therefore, it may be considered that the position of crystals in space is determined within errors of *0.4 mm. 3. Results and considerations 3.1. Falling velocity o f columnar snow crystals The observed columnar crystals were classified into the four types according to the manner as given by Magono and Lee (1966), namely hollow column, rimed needle, rimed sheath and rimed column.

4 October 1976 M. Kajikawa 279 Fig. 6 Axial ratio (l/d) of the various types of columnar crystals, l and d are the length along c-axis and a-axis of crystals, respectively. Marks of N., Z. & V. and L. & H. (the range of l/d) represent the observational results of Nakaya (1954), Zikmunda and Vali (1972) and Locatelli and Hobbs (1974), Fig. 7 Falling velocity (v) vs. length (l) along c-axis of columnar crystals. N. & T., Z. & V. and L. & H. represent the observed values of Nakaya and Terada (1935), Zikmunda and Vali (1972) and Locatelli and Hobbs (1974), respectively. Solid lines show the mean values of velocities for the four types of columnar crystals. Fig. 6 shows the axial ratio (l/d) of columnar crystals. In the case of rimed crystals, the length (l) and diameter (d) were both taken as average from this table the larger the length of a-axis is values over the rimed contours of crystals. The the greater the falling velocity is for the crystals observed values by Zikmunda and Vali (1972) of similar length of c-axis. This tendency has are approximately similar to the present results been assumed from the laboratory experiment of rimed sheath and rimed column, on the other related to the falling velocity of columnar snow hand the observed values of rimed column by crystals (for example, Jayaweera and Mason, Locatelli and Hobbs (1974) are slightly small, in general. 1965; Jayaweera 1971). and Cottis, 1969; Kajikawa, The falling velocity (the vertical component The relationship between the falling velocity of falling motion) of hollow column is summarized (v) and the length (l) of columnar crystals is in Table 1. The calculated values in this shown in Fig. 7. The solid lines are mean values table were obtained by the same method as Kajikawa of present measurements for the four types. The (1972), using the drag coefficient of circular broken lines are observed values by Locatelli cylinder with the same axial ratio and mass. and Hobbs (1974) for the rimed columns and These calculated values are close approximation by Nakaya and Terada (1935) for the needle type to the observed velocities. Moreover, it is seen crystals. Because of the scatter of axial ratio Table 1. Falling velocity of hollow columns.

5 280 Journal of the Meteorological Society of Japan Vol. 54, No. 5 Fig. 8 Relationship between bulk density (6) and length along c-axis (1) of columnar crystals. The marks are identical with Fig. 6. (Fig. 6) and bulk density (Fig. 8), the observed velocities are also dispersed. The observed values of velocity by Locatelli and Hobbs (1974) are greater than that of present measurement. The main reason of this discrepancy may be ascribed to the difference of axial ratio (in other words, the difference of external form), since the difference between the both observed values for the mass of rimed columns are not so large as seen in Fig. 9. It is considered that l/d of their observation is smaller than that of present observation (see Fig. 6) and those crystals observed by them are analogous to prolate spheroids consequently, so that the velocity of their observation is greater than present values. Moreover, the velocity of rimed needles was slightly smaller than that of needles observed by Nakaya and Terada (1935), although l/d (Fig. 6) of needles was larger and 6 (Fig. 8) of them was slightly smaller, than those of rimed needles. The factors affecting it seem to be the increase of drag coefficient due to the surface roughness and Fig. 9 Experimental formula of the mass (m) of columnar crystals. The marks are identical with Fig. 7. the decrease of ventilation in the both ends of rimed needles, because the porous ends of them are filled up by cloud droplets in riming process gradually. The dotted line in Fig. 7 is the calculated value of falling velocity for rimed needles, by the same method as hollow columns. It is considered that the observed values are slightly smaller than the calculated value, because of the increase of drag coefficient due to surface roughness of rimed crystals on comparing to the circular cylinder of model. Fig. 10 is the experimental formula of falling velocity of columnar crystals. Fig. 10 Experimental formula of the falling velocity (v) of columnar crystals. The marks are identical with Fig. 6.

6 October 1976 M. Kajikawa Free- f all pattern o f columnar snow crystals All of the rimed crystals observed in this study exhibited the rotation about vertical axis (namely, the change of *5 in Fig. 4) and the oscillation in vertical plane (the change of *), at the same time during fall. In addition to these unstable motions, the rimed needles and the rimed sheaths exhibited the horizontal movement apart from the vertical line during fall. However, the rotation about c-axis, which was observed by Zikmunda and Vali (1972), could not be detected. The existence of unstable motion of hollow columns could not be expressed clearly, but this problem will be discussed at a later date. The horizontal movement (projection of falling motion on the horizontal plane) of center of the rimed needle (Fig. 2) is shown in Fig. 11. In this figure, open circles and number indicate the positions taken in time interval of 1/100 sec, corresponding to the stereoscopic photographs of Fig. 2. A detailed analysis of fall pattern of this crystal is presented in Fig This figure illustrates that the continuous rotation (the change of *) about vertical axis and the oscillation (the change of *) of c-axis in vertical plane. Accordingly, the trajectory of three-dimensional movement of this crystal seems to be a spiral with slightly inclined axis. It can be seen from Fig. 12 that the period (T) of rotation is about 0.27 sec and that of oscillation is approximately one-half of it. The fluctuation of falling velocity was small, Fig. 11 Horizontal movement of the rimed needle of Case A (Fig. 2). a is the amplitude of spiral movement (the radius of spiral trajectory in falling motion). although the considerable change of * was in existence. In the crystal of Case B, its trajectory of fall motion was vertical as seen in Fig. 3. The fall pattern and the photomicrograph of this crystal (rimed sheath) are shown in Fig. 13. The period of continuous rotation is about 0.11 sec and that Fig. 12 Detailed analysis of fall pattern of Case A. The number of abscissa is corresponding to the images on stereophotographs of Fig. 2. T is the period of continuous rotation about vertical axis.

7 282 Journal of the Meteorological Society of Japan Vol. 54, No. 5 Fig. 14 Relationship between Reynolds number (Re) and non-dimension frequency (nd/v) of the continuous rotation. The period (T=1 /n) of rotation was distributed in the range 0.1 to 0.5 sec, for 90% of the observed rimed columnar crystals. Fig. 13 Detailed analysis of fall pattern for the rimed sheath of Case B. of oscillation is approximately one-half of it, being analogous to Case A. The fluctuation of velocity was corresponding to the change of *, namely the larger * the greater v. Fig. 14 is the relationship between Reynolds number (Re = vd/ v, where v is the falling velocity, d the length of a-axis or the diameter of columnar crystals and v the kinematic viscosity of air) and non-dimension frequency (nd/v, where n is the frequency of rotation). The nondimensional frequency of rotation increases in proportion to Re. From the laboratory model experiments (for example, Jayaweera and Mason, 1965; Podzimek, 1965, 1968) it is known that the shedding of Karman vortex at Re>50 causes the circular cylinders of model crystals to oscillate, when they fall through liquid. However, in the range of Re>5 are rimed columnar crystals exhibited the continuous rotation about vertical axis with the oscillation in vertical plane, as seen in Fig. 14. It is considered that the main reason of this discrepancy between the model experiment and the observation is due to the uneven distribution of mass of rimed columnar crystals. In connection to this matter, the influence of captured cloud droplets on the orientation and falling motion was investigated by the laboratory experiment of Jayaweera and Mason (1966). Fig. 15 Relationship between Re and nondimensional amplitude (2a/l) of spiral movement. The relationship between 2a/l (where a is the amplitude of spiral movement) and Re for the crystals fell with measurable spiral trajectories is shown in Fig. 15. All of the observed rimed columns fell with vertical trajectories in their centers, although the continuous rotation and oscillation are observed. The amplitude of spiral trajectory was measured from the projection of falling motion on horizontal plane (see Fig. 11), for the rimed needles and rimed sheaths of about 50%. In the case of remaining about 50%, the amplitude could not measure because of the projection of falling motion was considerably irregular. It can be seen from Fig. 15 that the non-

8 October 1976 M. Kajikawa 283 dimensional amplitude (2a/ l) decreases in inverse proportion to Re and the amplitude of spiral movement is very small at Re>50, for the rimed columnar snow crystals. 4. Concluding remarks The relationship between the falling velocity and the length of c-axis or mass was obtained for the various types of columnar snow crystals, as shown in Figs. 7 and 10. The falling velocity increases with the increase of their size or mass, in general. The remarkable difference of velocity is also found on the different types, at the same size of crystals. As a result of the comparison between the observed velocity and the calculated value, it appeared' that the falling velocity of hollow columns was well explained by the calculation method which use the drag coefficient determined by the model experiment related to the falling columnar crystals. The fall pattern of rimed columnar crystals was analysed by the stereophotogrammetric method. All of the rimed crystals exhibited the rotation about vertical axis and the oscillation in vertical plane, during fall. The crystals of about 50% in the rimed needles and sheaths fell with clear spiral trajectories, although the rimed columns fell with vertical paths. The period of oscillation was about one-half that of rotation and the non-dimensional frequency of rotation increased in proportion to Reynolds number of falling crystals. Acknowledgments The author wishes to thank Prof. C. Magono, Hokkaido University, for making the Mt. Teine observatory available for this study. The expense of this work was defrayed from a fund for scientific research of Education Ministry of Japan. References Jayaweera, K. 0. L. F., and B. J. Mason, 1965: The behaviour of freely falling cylinders and cones in a viscous fluid. J. Fluid Mech., 22, and, 1966: The falling motions of loaded cylinders and discs simulating snow crystals. Quart. J. Roy. Meteor. Soc., 92, and R. E. Cottis, 1969: Fall velocities of plate-like and columnar ice crystals. Quart. J. Roy. Meteor. Soc., 95, Kajikawa, M., 1971: A model experimental study on the falling velocity of ice crystals. J. Meteor. Soc. Japan, 49, : Measurement of falling velocity of individual snow crystals. J. Meteor. Soc. Japan, 50, : Experimental formula of falling velocity of snow crystals. J. Meteor. Soc. Japan, 53, Locatelli, J. D., and P. V. Hobbs, 1974: Fall speeds and masses of solid precipitation particles. J. Geophys. Res., 79, Magono, C., and C. W. Lee, 1966: Meteorological classification of natural snow crystals. J. Fac. Sri. Hokkaido Univ., Ser. VII, 2, Nakaya, U., and T. Terada, Jr., 1935: Simultaneous observations of the mass, falling velocity and form of individual snow crystals. J. Fac. Sci. Hokkaido Imp. Univ., Ser. II, 1, : Snow Crystals, natural and artificial. Harvard Univ. Press, 510 pp. Podzimek, J., 1965: Movement of ice particles in the atmosphere. Proc. Intern. Con f. Cloud Physics, Tokyo and Sapporo, : Aerodynamic conditions of ice crystal aggregation. Proc. Intern. Con f. Cloud Physics, Toronto, Sasyo, Y., 1971: Study of the formation of precipitation by the aggregation of snow particles and the accretion of cloud droplets on snowflakes. Pap. Meteor. Geophys., 22, Zikmunda, J., and G. Vali, 1972: Fall patterns and fall velocities of rimed ice crystals. J. Atmos. Sci., 29,

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