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  • View in gallery

    Example of the dynamic mesh technique applied to two randomly selected snapshots of a 5-mm broad-branched (P1c) crystal, demonstrating that the mesh adapts with time to match varying orientations of the crystal.

  • View in gallery

    Pressure deviation (Pa) and streamlines (2D projection onto the central yz plane) around (left) 2-mm P1b and (right) 3-mm P1c planar crystals with Reynolds number near 150. Pressure is shown by color shades (red: positive; blue: negative) and contoured (solid: positive; dashed: negative) over a range from −0.35 to 0.35. All snapshots correspond to the same randomly selected time step.

  • View in gallery

    Pressure deviation (shaded; Pa) around the edge of a 1-mm P1b with Reynolds number near 40 in the central yz plane. The time of the snapshot is as in Fig. 2. A few streamtraces (white) on this cross section are also shown.

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    Pressure deviation (shaded; Pa) on the (a) top, (b) bottom, and (c) side surfaces of a 2-mm broad-branched crystal (P1c) with Reynolds number near 95. The time of the snapshot is as in Fig. 2.

  • View in gallery

    As in Fig. 2, but for (left) 5-mm broad-branched crystal (P1c) and (right) 5-mm sector plate (P1b) planar crystals with Reynolds number near 350. Pressure is shown by color shades (red: positive; blue: negative) and contoured (solid: positive; dashed: negative) over a range from −0.60 to 0.60.

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    Dimensionless displacement, vibration frequencies, and characteristic angles for the planar ice crystals investigated in this study. The dimensionless displacement describes the characteristic length of the horizontal motions relative to the largest dimension of the ice crystal. The vibration frequency is defined as the frequency of the periodic behavior. The characteristic Tait–Bryan angles φ, θ, and ψ correspond to rotation about the x, y, and z axes of the crystal, respectively, and describe the orientation of the ice crystals with respect to their initial orientation. The symbols are as follows: * indicates that amplitude shows an increasing trend with time and may be underestimated, † indicates that the vibration frequency about the x axis is slightly larger than the y axis, ‡ indicates that amplitude shows a decreasing trend with time and may be overestimated, and § indicates that the vibration frequency about the y axis is slightly larger than the x axis.

  • View in gallery

    Fall trajectories of planar ice crystals as shown through consecutive snapshots of the crystal position on the xz plane for (a) sector plate (P1b), Re = 384, and (b) broad-branched crystal (P1c), Re = 345. The time interval is 0.005 s. The initial (final) position of the crystal is highlighted in green (red) on the projection.

  • View in gallery

    Vorticity distribution (s−1) around (left) an ordinary dendritic crystal (P1e) and (right) a sector plate (P1b). Vorticity magnitude is shown by color shades (yellow: low; bright blue: high) over a range of 100–4000 s−1. The snapshots are from randomly selected time steps.

  • View in gallery

    The 3D -vorticity isosurface (s−1) around a 5-mm sector plate (P1b) to illustrate the unsteady vortex structure. The blue isosurfaces correspond to -vorticity values of −400 s−1, while the red isosurfaces correspond to -vorticity values of 400 s−1. The snapshot corresponds to the same randomly selected time step, as in Fig. 2.

  • View in gallery

    The velocity (m s−1) and velocity vector (2D projection of the 3D vector onto the central yz plane) around a (left) broad-branched (P1c) and (right) stellar crystal (P1d) . The velocity is shown by color shades (red: positive; blue: negative) over a range from −0.40 to 0.40 m s−1. The snapshots are from randomly selected time steps.

  • View in gallery

    Terminal velocities (cm s−1) of planar crystals for 900 hPa. Square, circle, diamond, and triangle markers indicate the results for sector plates (P1b), broad-branched crystals (P1c), stellar crystals (P1d), and ordinary dendritic crystals (P1e), respectively. The corresponding thick curves are the power-law fits given by Eqs. (8)(11): solid curve for sector plates, dashed curve for broad-branched crystals, dotted curve for sector plates, and dash–dotted curve for dendrites. Thick, gray solid curve represents the 1000-hPa parameterization of hexagonal plates (P1a) from Cheng et al. (2015). The corresponding 1000-hPa terminal velocity parameterizations of Heymsfield and Kajikawa (1987) are shown by thin curves, with colors and line styles matching these experimental results.

  • View in gallery

    Drag coefficient of planar crystals. Square, circle, diamond, and triangle markers indicate the results for sector plates (P1b), broad-branched crystals (P1c), stellar crystal (P1d), and ordinary dendritic crystals (P1e), respectively. The corresponding curves are the power-law fits given by Eqs. (16)(19): solid curve for sector plates, dashed curve for broad-branched crystals, dotted curve for sector plates, and dash–dotted curve for dendrites. Bolded bullets represent the steady-state drag coefficients for hexagonal plates (P1a) from Hashino et al. (2014). Thick gray solid curve represents the parameterization of hexagonal plates from Cheng et al. (2015). The parameterizations of Wang and Ji (1997) are shown in thin lines: solid curve for hexagonal plates and dashed curve for broad-branched crystals. The adjusted Cd values discussed in Heymsfield and Westbrook (2010) are shown as the filled-in markers, colors and shapes corresponding to the crystal type described in the legend.

  • View in gallery

    Relation between the Reynolds number and Best number for the crystals studied here as well as those by Böhm (1989) and Heymsfield and Westbrook (2010).

  • View in gallery

    Vapor density distributions (g m−3) around planar crystals with maximum dimension (diameter) of 5 mm. Vapor density is shown by shades of blue (darker: approaching 2% supersaturation ρυ = 2.4086 g m−3; lighter: approaching saturation ρυ = 2.3613 g m−3). All snapshots correspond to the same randomly selected time step, as in Fig. 2.

  • View in gallery

    Ventilation coefficient of planar crystals. Square, circle, diamond, and triangle markers indicate the results for sector plates (P1b), broad-branched crystals (P1c), stellar crystals (P1d), and ordinary dendritic crystals (P1e), respectively. The corresponding curves are the power-law fits given by Eqs. (20)(23): solid curve for sector plates, dashed curve for broad-branched crystals, dotted curve for sector plates, and dash–dotted curve for dendrites. The parameterizations of Ji and Wang (1999) are shown in thin lines: dotted curve for circular columns, solid curve for hexagonal plates, and dashed curve for broad-branched crystals. The general expression for the ventilation coefficient based on data for spheres and oblate spheroids from Hall and Pruppacher (1976) is shown in the thin black dash–dotted line.

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A Numerical Study on the Aerodynamics of Freely Falling Planar Ice Crystals

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  • 1 Department of Atmospheric and Oceanic Sciences, University of Wisconsin–Madison, Madison, Wisconsin
  • 2 Department of Atmospheric and Oceanic Sciences, University of Wisconsin–Madison, Madison, Wisconsin, and Research Center for Environmental Changes, Academia Sinica, Taipei, Taiwan
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Abstract

Fluid flow fields and fall patterns of falling planar ice crystals are studied by numerically solving the unsteady, incompressible Navier–Stokes equations using a commercially available computational fluid dynamics package. The ice crystal movement and orientation are explicitly simulated based on hydrodynamic forces and torques representing the 6 degrees of freedom. This study extends the current framework by investigating four planar-type ice crystals: crystals with sector-like branches, crystals with broad branches, stellar crystals, and ordinary dendritic crystals. The crystals range from 0.2 to 5 mm in maximum dimension, corresponding to Reynolds number ranges from 0.2 to 384. The results indicate that steady flow fields are generated for flows with Reynolds numbers less than 100; larger plates generate unsteady flow fields and exhibit horizontal translation, rotation, and oscillation. Empirical formulas for the drag coefficient, 900-hPa terminal velocity, and ventilation effect are given. Fall trajectory, pressure distribution, wake structure, vapor field, and vorticity field are examined.

Current affiliation: National Weather Service, Silver Spring, Maryland.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Prof. Pao K. Wang, pwang1@wisc.edu

Abstract

Fluid flow fields and fall patterns of falling planar ice crystals are studied by numerically solving the unsteady, incompressible Navier–Stokes equations using a commercially available computational fluid dynamics package. The ice crystal movement and orientation are explicitly simulated based on hydrodynamic forces and torques representing the 6 degrees of freedom. This study extends the current framework by investigating four planar-type ice crystals: crystals with sector-like branches, crystals with broad branches, stellar crystals, and ordinary dendritic crystals. The crystals range from 0.2 to 5 mm in maximum dimension, corresponding to Reynolds number ranges from 0.2 to 384. The results indicate that steady flow fields are generated for flows with Reynolds numbers less than 100; larger plates generate unsteady flow fields and exhibit horizontal translation, rotation, and oscillation. Empirical formulas for the drag coefficient, 900-hPa terminal velocity, and ventilation effect are given. Fall trajectory, pressure distribution, wake structure, vapor field, and vorticity field are examined.

Current affiliation: National Weather Service, Silver Spring, Maryland.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Prof. Pao K. Wang, pwang1@wisc.edu
Keywords: Clouds; Ice crystals

1. Introduction

Consider a tiny, fragile ice crystal falling before you, drifting gently toward the surface of Earth. This is simply one of many beautiful, everyday examples of time-dependent fluid dynamics at low to intermediate Reynolds numbers. When envisioning the fall trajectory of a snow or ice crystal, firsthand experience, or perhaps a bit of imagination, tells us to expect complex translational and rotational motions, such as side-to-side oscillations (fluttering), sideways drifting, and tumbling. The goal of this study is to extend the framework laid out in Cheng et al. (2015) by obtaining the theoretical numerical solutions of the flow fields around four types of planar ice crystals—crystals with sector-like branches (P1b), crystals with broad branches (P1c), stellar crystals (P1d), and ordinary dendrites (P1e)—of various sizes, where smaller sizes exhibit steady flow fields and larger sizes demonstrate unsteady flow fields with eddy shedding.

Frozen hydrometeors take on all types of shapes and sizes, from hexagonal ice crystal plates to more complicated dendrites and aggregates. Ice crystals of various shapes and sizes have varying masses and cross-sectional areas, thus generating different flow fields, which influence the microphysics of the particles, including their fall velocities, heat dissipation rates, ventilation effects, diffusional growth rates, and collision efficiency with other hydrometeors (Pruppacher and Klett 1997; Wang 1982; Wang and Denzer 1983). Crystal and snow processes influence thunderstorm anvil and other cloud structures, which have important radiative impacts (e.g., Takano and Liou 1993). Quantitative knowledge of all of these microphysical processes is required for accurate weather and climate predictions.

Flow fields and fall attitudes can be studied experimentally from field observations of real ice crystals falling through air (e.g., Nakaya and Terada 1935), tank measurements of model ice crystals falling through viscous liquids (e.g., Jayaweera and Mason 1965, 1966; List and Schemenauer 1971; Willmarth et al. 1964), or from theoretical calculations (e.g., Wang 2001; Cheng et al. 2015; Hashino et al. 2016, 2014; Ji and Wang 1991; Wang and Ji 1997, 2000). More recent studies such as Matrosov et al. (2005) have inferred fall attitudes of ice crystals from radar data, and Takahashi (2014) has used a cloud tunnel to investigate growth processes. Observational studies cover a broad range of natural crystal shapes, though these measurements are subject to large experimental error, while tank experiments have small experimental errors (~10%), but only a small sampling of idealized crystal shapes are studied (Heymsfield and Westbrook 2010). Theoretical calculations are essential for fully understanding the motions of ice crystals and making meaningful assessments of their role in cloud microphysical processes.

Previous studies have found that small ice crystals exhibit steady fall behaviors, though very small particles [Reynolds number (Re) ≤ 1] likely fall with random orientations (Wang 2013). Planar hexagonal ice crystals in the intermediate Reynolds range (1 ≤ Re ≲ 100) fall steadily with horizontally oriented basal planes (Bréon and Dubrulle 2004). The upper Reynolds limit for steady motion for planar ice crystals with narrow branches, such as stellar crystals (P1d) and dendrites (P1e), is Re ≈ 200. Note that these Reynolds ranges are not exact, and different crystal habits, such as ice columns, have different Reynolds ranges that describe steady fall behavior. The flow fields become unsteady and downstream eddy shedding begins to occur for increasingly larger ice crystals. Eventually, secondary motions occur simultaneously, such as rotational, oscillatory, and weaving translational motions that are easy to envision though complicated to understand.

The use of numerical methods for studying flow fields around particles has a long history. For example, Dennis and Chang (1969), Kawaguti (1953), Nieuwstadt and Keller (1973), and Thom (1933) studied the two-dimensional flow past circular cylinders, and Masliyah and Epstein (1970) and Rimon and Lugt (1969) studied flow around thin, axisymmetric oblate spheroids. Often, early studies made assumptions and approximations that are not realistic. For example, Schlamp et al. (1975) assumed falling columnar ice crystals could be approximated as infinitely long circular cylinders, reducing the three-dimensional problem to one in two dimensions. Similarly, Pitter et al. (1973) approximated hexagonal ice crystals as thin oblate spheroids.

Ji and Wang (1990, 1991) and Wang and Ji (1997) used realistic columnar and hexagonal plate ice crystal shapes in their flow field calculations, obtaining steady and unsteady flow fields for low to intermediate Reynolds numbers. One limitation of these studies, and all prior studies, is the assumption that the ice crystals fall with their largest dimension (length axis for columnar crystals; basal plane for hexagonal plates) oriented perpendicular to the direction of fall, which is the expected orientation of steady falling ice crystals (Pruppacher and Klett 1997). Of course, firsthand experience tells us that snow and ice crystals do not fall down straight with no changes in orientation; rather, the motion depends on the Reynolds number and dimensionless moment of inertia (e.g., Field and Klaus 1997; Willmarth et al. 1964). Unstable fall behavior has been shown to occur for platelike crystals as small as 1.23 mm, corresponding to Re = 47, and the orientation and horizontal motions are important in crystal aggregation (Kajikawa 1992).

The first numerical study of flow past inclined ice crystals was performed by Hashino et al. (2014), which provided insight into the flow fields of ice crystals with fixed inclined orientation of the crystal’s largest dimension, but even these results would differ from the realistic case of unsteady freely falling crystals. Up to this point, by not simulating freely falling ice crystals, the fall motions generate flow fields, but the crystals themselves do not respond to changes in the flow fields. Cheng et al. (2015) addresses this deficiency and provides the first numerical results for the flow-field calculation of freely falling hexagonal ice plates, allowing for the hydrodynamic forces of the flow field to influence the plate, such that oscillatory, rotational, and translational motions are allowed. In a similar manner, Hashino et al. (2016) simulates the flow fields around freely falling columnar crystals.

In addition to calculating flow fields, previous theoretical studies have also investigated the enhancement of ice crystal diffusional growth due to falling motion, or the ventilation effect (e.g., Cheng et al. 2014; Hall and Pruppacher 1976; Ji and Wang 1999; Masliyah and Epstein 1970; Pitter et al. 1974). Earlier studies approximated hexagonal plates as thin oblate spheroids, though with the advance of computer technology, it is possible to perform calculations using the true shapes of ice plates. Similar to Cheng et al. (2014), in this study, specific water vapor density boundary conditions are prescribed in order to calculate the ventilation effect, or the enhancement in growth by vapor diffusion due to falling motion.

Working to extend the framework laid out by Cheng et al. (2015) and previous studies, the flow fields of four additional types of planar ice crystals—crystals with sector-like branches (P1b), crystals with broad branches (P1c), stellar crystals (P1d), and ordinary dendrites (P1e)—are calculated for crystal size ranges from 0.2 to 5 mm in maximum dimension. The methods of this study are given in the following section. Section 3 presents the results and discussion. Conclusions and outlooks are given in section 4.

2. Mathematics and physics of the flow field calculation

In this study, theoretical calculations are performed to explicitly simulate falling planar ice crystals based on the forces and torques acting on them from the environmental flow field. The plates exhibit translational and rotational motions corresponding to all 6 degrees of freedom using a similar technique to that of Cheng et al. (2015).

Planar ice crystals come with a multitude of habits, and it is impossible to study the flow fields of all of them. This study investigates the flow fields around crystals with sector-like branches (P1b), crystals with broad branches (P1c), stellar crystals (P1d), and ordinary dendrites (P1e), in accordance with the Magono and Lee (1966) natural snow crystal classification scheme. These four varieties are more complicated than those studied by previous investigators, and we hope to use them to understand how these added complexities in shape influence the flow field and ventilation coefficient. Crystal diameter and thickness are determined using empirical power-law relationships given by Auer and Veal (1970). The crystal density ρs is assumed to be 916.68 kg m−3. The plates are assumed to fall in an atmospheric environment of air density ρa = 1.19 kg m−3 and kinematic viscosity ν = 1.4019 × 10−5 m2 s−1 (approximating environmental conditions 900 hPa and −10°C). Further, to understand the effect of flow on diffusional growth, the crystals are assumed to fall through a supersaturated column such that the air is saturated at the surface of the crystal (i.e., 100% relative humidity with respect to water), and the environment is 2% supersaturated sufficiently far from the crystal (i.e., 102% relative humidity with respect to water). This results in vapor density ρυ boundary conditions: ρυ = 2.3613 × 10−3 kg m−3 at the crystal surface and ρυ = 2.4086 × 10−3 kg m−3 at the outer boundary.

The time-dependent Navier–Stokes equation coupled with the incompressible flow condition is the governing set of equations for the flow field around falling ice crystals (Pruppacher and Klett 1997):
e1
e2
where u is the air velocity, p is the static pressure, ν is the kinematic viscosity, and g is the gravitational acceleration, a constant at −9.81 m s−2. The boundary conditions are
e3
e4
e5
where is the reference wind speed (close to the terminal fall velocity of the ice crystal) and ez is the unit vector along the fall direction. These are the common boundary conditions for these types of problems (Wang 2013).

The computational fluid dynamics package Fluent 15.0.0 and 16.2.0 of ANSYS, Inc., are used for simulating the flow fields around falling planar ice crystals. A coupled pressure-based solver is used, which involves simultaneously solving for the momentum equations and the pressure-based continuity equation in a closely coupled manner. The pressure implicit with splitting of operators (PISO) algorithm is chosen for solving the pressure–velocity coupling (Issa 1986). The body force–weighted pressure interpolation scheme is chosen for interpolating pressure values since the body forces are known a priori in the momentum equations. The spatial discretization of the advection of momentum is solved with the Quadratic Upstream Interpolation for Convective Kinematics (QUICK) scheme (Freitas et al. 1985). A second-order upwind scheme is chosen for the energy discretization, which is used for the calculation of vapor diffusion, and provides high-order accuracy over other methods. A first-order implicit scheme was chosen for the transient formulation with a time step typically set to 1 × 10−4 s, though some simulations required smaller time steps in order to better resolve large displacements because of the forces exerted on the crystal from the flow. For more information on the computational fluid dynamics package used, please refer to the ANSYS Fluent technical report (ANSYS 2013).

Similar to Cheng et al. (2015), the three-dimensional computational domain is discretized with an unstructured tetrahedral grid, which allows precise fitting to the ice crystal shape. The domain consists of about 1.5–2.5 × 106 cells (more for larger diameter/smaller aspect ratio). The mesh is finer toward the crystal surface and in the crystal wake and becomes coarser farther away from the crystal and wake. ANSYS Fluent provides dynamic mesh capabilities, which allows the mesh to adapt and match the position and orientation of the crystal at each time step. Figure 1 illustrates the dynamic mesh technique.

Fig. 1.
Fig. 1.

Example of the dynamic mesh technique applied to two randomly selected snapshots of a 5-mm broad-branched (P1c) crystal, demonstrating that the mesh adapts with time to match varying orientations of the crystal.

Citation: Journal of the Atmospheric Sciences 75, 9; 10.1175/JAS-D-18-0041.1

3. Results and discussion

We report the results of falling ice crystals with diameters (maximum dimension) 0.2, 0.3, 0.4, 0.5, 1, 2, 3, 4, and 5 mm. The ice plates are not assumed stationary initially; rather, they fall at a reference velocity close to the terminal velocity. Velocity power-law relationships from Mitchell and Heymsfield (2005) are used for determining a first guess terminal velocity. Most simulations were repeated several times using refined reference velocities to improve convergence and reduce boundary condition influences.

Flow fields, fall behavior, velocity, and vapor fields are discussed. While there are 36 cases—4 crystal habits and 9 diameters—simulated, the results are comparable across different crystal sizes, and thus, only a select number of cases are presented as a representation of the results.

a. Flow characteristics

Figure 2 shows the pressure distribution and streamlines around two planar ice crystals with a Reynolds number near 150 (range 115–175, corresponding to the 2-mm sector plate (P1b) and 3-mm broad-branched crystal (P1c) on the yz plane for a randomly selected time step consistent across each panel. The Reynolds number is defined as
e6
where the diameter d used is that of the maximum dimension of the crystal, u is the terminal velocity, and ν is the kinematic viscosity. The flows are steady in each case, with a general left–right symmetry about the central x axis despite slight rotation about the z axis in many cases. The region of relatively higher pressure in the upstream region is expected and may work to prevent tiny crystals, droplets, or aerosol particles from colliding with the crystal (Vittori and Prodi 1967). Standing eddies are present in the wake of the sector plate (P1b) and broad-branched crystal (P1c); the ice plates are falling vertically, while the characteristics of the eddies remain constant with time. The ice crystals mainly affect the flow within the vertical column in which they fall, only slightly disturbing the area outside this column up to a distance of a few diameters. The results are consistent across the current study for crystals with Re near and less than 100 and are in general agreement with previous studies (e.g., Cheng et al. 2015; Willmarth et al. 1964).
Fig. 2.
Fig. 2.

Pressure deviation (Pa) and streamlines (2D projection onto the central yz plane) around (left) 2-mm P1b and (right) 3-mm P1c planar crystals with Reynolds number near 150. Pressure is shown by color shades (red: positive; blue: negative) and contoured (solid: positive; dashed: negative) over a range from −0.35 to 0.35. All snapshots correspond to the same randomly selected time step.

Citation: Journal of the Atmospheric Sciences 75, 9; 10.1175/JAS-D-18-0041.1

While negative deviations in pressure generally occur in the downstream wake, pressure minima also occur at the upper edge of the plate (Fig. 3). This behavior is consistent with other numerical studies that treat planar ice crystals as hexagonal plates with finite thickness (two sharp edges), such as Cheng et al. (2015), Hashino et al. (2014), Ji and Wang (1991), and Wang and Ji (1997). This configuration of the pressure field at the upper edge of an ice plate likely impacts the collision efficiency, altering the riming rate and the location at which a supercooled drop will strike a plate. This pressure configuration is consistent across all plates in this study.

Fig. 3.
Fig. 3.

Pressure deviation (shaded; Pa) around the edge of a 1-mm P1b with Reynolds number near 40 in the central yz plane. The time of the snapshot is as in Fig. 2. A few streamtraces (white) on this cross section are also shown.

Citation: Journal of the Atmospheric Sciences 75, 9; 10.1175/JAS-D-18-0041.1

A look at the pressure distribution over the surfaces of a 2-mm P1c is shown in Fig. 4. Both basal planes are shown as viewed from above and below, along with the edges of the crystal as viewed from the side. This perspective of the low pressure side shines light on the location of the pressure minima along the edges seen in the yz-plane slices. Notice that there is not a uniform region of pressure minima along the perimeter of the crystal; rather, the minima occur at the peaks of the branches of the crystal and taper off toward the center. This configuration works to explain the process of growth along branches, reproducible in modeling studies (e.g., Gravner and Griffeath 2009; Pitter and Pruppacher 1974). The pressure distribution on the surface of the high pressure side of the crystal is more chaotic than the other side. An extremely small sliver of low to negative pressure is observed along the entire perimeter of the branches. Finally, the side view echoes the observations from the basal surfaces; the pressure distribution is such that the minima occur at the peaks, with pressure increasing toward the center of the crystal.

Fig. 4.
Fig. 4.

Pressure deviation (shaded; Pa) on the (a) top, (b) bottom, and (c) side surfaces of a 2-mm broad-branched crystal (P1c) with Reynolds number near 95. The time of the snapshot is as in Fig. 2.

Citation: Journal of the Atmospheric Sciences 75, 9; 10.1175/JAS-D-18-0041.1

As expected, the flow becomes unsteady with increasing Reynolds number (Fig. 5). The stellar crystal (P1d) and ordinary dendritic crystals (P1e) do not exhibit unsteady fall behavior over any range of Reynolds number simulated in this study; the sector plate (P1b) and broad-branched crystal (P1c) begin to demonstrate unsteady behavior for diameters of 3 mm and above for the sector plate and 4 mm and above for the broad-branched crystal, corresponding to the Reynolds numbers 198 and 258, respectively. Snapshots of a 5-mm-diameter sector plate and broad-branched crystal are presented in Fig. 5. The upstream high pressure region, along with pressure minima along the edge of the crystal surface, are two similarities between the steady cases. The eddies in the downstream wake are no longer symmetric, with eddy shedding occurring in the downstream. Unlike the steady cases in Fig. 2, the pressure minimum in the wake is not always in contact with the surface of the plate but is found slightly above the surface or in contact with only one region. The pressure minimum changes in both magnitude and location on a near-periodic basis, resulting in eddy shedding. Consistent with Cheng et al. (2015), the location of the upstream pressure maximum is not stationary either but tends to vary with the orientation of the crystal. As the crystal tilts in a direction, the asymmetric maximum pressure distribution on the underside of the plate will generate a torque, causing the crystal to begin to take on the opposite inclination and move in the opposite direction.

Fig. 5.
Fig. 5.

As in Fig. 2, but for (left) 5-mm broad-branched crystal (P1c) and (right) 5-mm sector plate (P1b) planar crystals with Reynolds number near 350. Pressure is shown by color shades (red: positive; blue: negative) and contoured (solid: positive; dashed: negative) over a range from −0.60 to 0.60.

Citation: Journal of the Atmospheric Sciences 75, 9; 10.1175/JAS-D-18-0041.1

b. Fall attitudes

An analysis of the dimensionless displacement, vibration frequencies, and characteristic angles is performed, and the results of each crystal type are presented in Fig. 6. Results are omitted when the vibration frequency or characteristic angles are not obvious or would require subjective interpretation. The vibration frequency (frequency of the periodic behavior) about the x and y axis are generally the same, though it is noted when the frequencies differ slightly, and both frequencies are listed for 0.5-mm ordinary dendritic crystal (P1e), which differed significantly.

Fig. 6.
Fig. 6.

Dimensionless displacement, vibration frequencies, and characteristic angles for the planar ice crystals investigated in this study. The dimensionless displacement describes the characteristic length of the horizontal motions relative to the largest dimension of the ice crystal. The vibration frequency is defined as the frequency of the periodic behavior. The characteristic Tait–Bryan angles φ, θ, and ψ correspond to rotation about the x, y, and z axes of the crystal, respectively, and describe the orientation of the ice crystals with respect to their initial orientation. The symbols are as follows: * indicates that amplitude shows an increasing trend with time and may be underestimated, † indicates that the vibration frequency about the x axis is slightly larger than the y axis, ‡ indicates that amplitude shows a decreasing trend with time and may be overestimated, and § indicates that the vibration frequency about the y axis is slightly larger than the x axis.

Citation: Journal of the Atmospheric Sciences 75, 9; 10.1175/JAS-D-18-0041.1

The dimensionless horizontal displacement l* describes the characteristic length of the horizontal motions relative to the largest dimension of the ice crystal, defined as
e7
where l is the length of swing, spiral, and irregular motions of the crystal. The dimensionless horizontal displacement of the crystals ranges from 0.009 to 3.525. In other words, the falling crystals simulated in this study move horizontally with the smallest (largest) distance of 0.9% (353%) of its diameter. The crystal type with the largest range in l* is the sector plate (P1b); the smallest range in l* is seen with the stellar crystal (P1d). As in Cheng et al. (2015), no systematic relationship appears to exist between diameter and horizontal dimensionless displacement.

The horizontal motions and orientation of ice crystals may have impacts on the collision efficiency of two crystals. Currently, collision efficiency is defined under the assumption that a collector hydrometeor falls only vertically, ignoring horizontal motions (Pruppacher and Klett 1997; Wang 2013). An ice crystal falling with a horizontal velocity component will travel a farther distance than one falling with pure vertical motion, providing more opportunity to collide with small droplets at a different efficiency than the vertically falling plate. This is a complicated matter, and more studies will be conducted in the future. Crystal orientation is important when considering radar backscatter signals from ice plates (e.g., Ishimoto 2008; Matrosov 2007).

The Tait–Bryan angles φ, θ, and ψ are used to quantitatively describe the orientation of the ice crystals with respect to their initial orientation (Fig. 6). The angles φ, θ, and ψ are defined and correspond to rotation about the x, y, and z axes of the crystal, respectively. The characteristic angle and vibration frequency results are confirmed using supporting evidence from performing a fast Fourier transform (FFT) of the orientation data. The vibration frequency about the z axis has been omitted when the high-frequency signal was not obvious compared to the low-frequency signal.

Rotation about the z axis is observed with the small, steady-state crystals, though the characteristic angles are generally ordered at 10−1° or 10−2°. Nothing in principle should cause spinning in the steady cases of the idealized crystals. This behavior likely arises from slight imperfections in the meshing processes, which causes inevitable asymmetry. Lack of exact symmetry of the mesh and crystal, leading to an uneven distribution of mass, is a possible explanation for some crystal cases demonstrating oscillation about a preferential basal face-parallel axis (a axis) even though the moments of inertia about any a axis are the same.

The xz plane trajectories for crystals with Reynolds number near 350 are shown in Fig. 7. The starting positions, with no initial inclination angle, are shown by the projections highlighted in green; the end positions, occurring at 0.5 s, are shown by the red highlighted projections. The projections are displayed every 0.005 s. These cases become less stable with time and appear quite unstable toward the end of the simulation time. The zigzag swing oscillations in the sector plate (P1b) result in horizontal translations of roughly 26% of the diameter of the plate; fluttering in the broad-branched crystal (P1c) produces a smaller horizontal displacement of 5% of its diameter. Intuitively, the crystal is displaced rightward (leftward) horizontally in response to tilting to the right (left), in agreement with experimental results of Willmarth et al. (1964) and Stringham et al. (1969) and consistent with the numerical results of Cheng et al. (2015) and Hashino et al. (2016).

Fig. 7.
Fig. 7.

Fall trajectories of planar ice crystals as shown through consecutive snapshots of the crystal position on the xz plane for (a) sector plate (P1b), Re = 384, and (b) broad-branched crystal (P1c), Re = 345. The time interval is 0.005 s. The initial (final) position of the crystal is highlighted in green (red) on the projection.

Citation: Journal of the Atmospheric Sciences 75, 9; 10.1175/JAS-D-18-0041.1

Crystal fall behavior and fall velocity are contributing factors for understanding the aggregation process of snow crystals (e.g., Sasyo 1971). An observational study by Kajikawa (1992) showed a relationship between increasing Reynolds number, above about 40, and unstable falling motion due to vortex shedding. In that study, unstable falling motion first began with oscillations about an a axis of platelike crystals then proceeded to display swinging motion, followed by rotation about the z axis, with increasing Re. Note, tumbling motions that were observed in tank experiments by Willmarth et al. (1964) and Stringham et al. (1969) were not observed in the Kajikawa (1992) observational study. Further, because of natural snow crystal asymmetry, the onset of observed unstable falling motion in the Kajikawa (1992) study occurred at considerably smaller Re values than in model experiments (e.g., List and Schemenauer 1971). In the current computational fluid dynamics (CFD) model experiment, the onset of unstable fall motion did not occur until higher Re, with the highest Re for stable motion occurring at Reynolds numbers 115 and 175 for the simulated sector plate (P1b) and broad-branched crystal (P1c), respectively; the stellar crystal (P1d) and ordinary dendritic crystal (P1e) exhibit stable falling motion for all Re considered in this study. For comparison, the highest Re for stable disk motion in the Willmarth et al. (1964) tank study was 172. In that study, tumbling motions only occurred for Re in excess of 2000, much larger Re than those observed in the current study. At this moment, we are unsure what causes this discrepancy because there are many differences between the dimension, shape, aspect ratio, and density of the crystals in this study and Willmarth et al.’s disks. To determine the discrepancy require further studies.

c. Vorticity

The structure of the wake and downstream flow, illustrated here using vorticity, may influence the collection of cloud droplets and scavenging of aerosol particles by ice crystals. The vorticity magnitude distributions on the cross section through the center of falling 5-mm sector plate (P1b; Re = 384) and ordinary dendritic crystal (P1e; Re = 173) are shown in Fig. 8. As expected, the maximum in the vorticity magnitude is observed at the edges of the crystal, where the flow is changing direction and speed. The vorticity is transported downstream by the flow, with higher vorticity forming a champagne-glass shape, with the center void of high vorticity for the sector plate. The vorticity is advected downstream for the dendrite as well, not only at the outer edges but at the interior edges as well. The vorticity is advected farther downstream in the 5-mm sector plate wake than the dendrite, because of the difference in the speed of the flow—terminal velocity of 1.08 and 0.49 m s−1 for the plate and dendrite, respectively. The interior of the wake in the sector plate contains a region of low vorticity (less than 100 s−1) located slightly above the surface of the crystal, appearing in a random pattern.

Fig. 8.
Fig. 8.

Vorticity distribution (s−1) around (left) an ordinary dendritic crystal (P1e) and (right) a sector plate (P1b). Vorticity magnitude is shown by color shades (yellow: low; bright blue: high) over a range of 100–4000 s−1. The snapshots are from randomly selected time steps.

Citation: Journal of the Atmospheric Sciences 75, 9; 10.1175/JAS-D-18-0041.1

The vorticities form intertwining vortex tubes in the downstream wake, shown in Fig. 9 as three-dimensional -vorticity isosurfaces of ±400 s−1. This structure may impact the collection of small cloud particles with subsequent ice crystals downstream by chaotically redistributing the particles in space. The positive–negative alternating pattern of the vorticity isosurfaces around the sector branches is intuitively understood considering the direction in which the flow will curl on either side of the branch.

Fig. 9.
Fig. 9.

The 3D -vorticity isosurface (s−1) around a 5-mm sector plate (P1b) to illustrate the unsteady vortex structure. The blue isosurfaces correspond to -vorticity values of −400 s−1, while the red isosurfaces correspond to -vorticity values of 400 s−1. The snapshot corresponds to the same randomly selected time step, as in Fig. 2.

Citation: Journal of the Atmospheric Sciences 75, 9; 10.1175/JAS-D-18-0041.1

The flow fields and fall attitudes discussed above are in general agreement with previous studies (e.g., Cheng et al. 2015; Willmarth et al. 1964). Features, such as the pressure minimum on the upper edge of the peaks of the ice plate, are prevalent in all cases and lend support to the observation that crystals tend to grow faster about their peaks. Projections of the fall attitudes of crystals at varying Reynolds number, along with an analysis of the characteristic angles, may be of use for the cloud modeling community.

d. Terminal velocity

Consistent with the findings of Cheng et al. (2015), the results of this study show that the terminal velocity of simulated ice crystals fluctuate throughout the fall process, generally bound within a certain range. The range is smaller for crystals that fall with the basal plane entirely normal to the fall direction, likely because the cross-sectional area exposed to the flow is nearly constant. The range is larger for crystals exhibiting swinging, unstable motion, because with an inclination relative to the flow, the implied smaller cross-sectional area decreases the upward drag force and increases the downward acceleration. The velocity relative to the flow initially varies from case to case as the crystal responds to the first-approximation base velocity to which it is subjected. For this reason, the terminal velocity is computed by averaging the last 3000 time steps of the simulation, as per the method outlined in Cheng et al. (2015).

Figure 10 shows the -velocity distribution on and velocity vectors around a 4-mm broad-branched crystal (P1c) and stellar crystal (P1d). Note the scales are panel specific, corresponding to the calculated 900-hPa terminal velocity for each crystal, 91.0 and 32.5 cm s−1, respectively. Note that the velocity vectors are projections of the 3D vectors onto the yz plane crossing the center of the plate; thus, not all vectors are on the same plane, and the uneven vector distribution is explained by the nonuniform mesh used to conform to the surface of the crystal and to allow for mesh adjustment with time. At distances sufficiently far from the crystal, the velocity nears the initial condition terminal velocity estimation at which the crystals are subjected during the simulation. Because of the no-slip condition, the fluid is stagnant everywhere on the solid crystal surface. Both crystals have areas of low velocity from the high pressure on the underside of the crystal (front stagnation region); the area of lower velocity is larger, relative to the diameter, for the broad-branched crystal than the stellar crystal (P1d) (Re = 260 and 89, respectively). The broad-branched crystal (P1c) displays a region of negative velocity, or return flow/recirculation, in the wake, as opposed to the stellar crystal, with lower Reynolds number. There is a degree of symmetry to both velocity fields, though the wake of the broad-branched crystal shows slight oscillation of the velocity magnitude at a distance of about 4 mm above the surface of the crystal. The velocity gradient near the crystal surfaces—and in the case of the 4-mm broad-branched crystal, in the wake as well—implies the vorticity will be large here.

Fig. 10.
Fig. 10.

The velocity (m s−1) and velocity vector (2D projection of the 3D vector onto the central yz plane) around a (left) broad-branched (P1c) and (right) stellar crystal (P1d) . The velocity is shown by color shades (red: positive; blue: negative) over a range from −0.40 to 0.40 m s−1. The snapshots are from randomly selected time steps.

Citation: Journal of the Atmospheric Sciences 75, 9; 10.1175/JAS-D-18-0041.1

Figure 11 presents the 900-hPa terminal velocity for each crystal type in this study alongside the numerical results of the 1000-hPa terminal velocity of the hexagonal plate P1a from Cheng et al. (2015) and the 1000-hPa observational terminal velocity parameterizations of Heymsfield and Kajikawa (1987) for the corresponding crystal type. The terminal velocity increases with increasing diameter, consistent with the observational data and previous numerical results. For the sector plates (P1b), broad-branched crystals (P1c), and ordinary dendritic crystals (P1e) investigated in this study, the 900-hPa terminal velocities are greater than those of the 1000-hPa observational ones, likely because the mass of the idealized crystals represent the upper bound for that of natural crystals, as discussed previously. The simulated stellar crystals (P1d) have calculated 900-hPa terminal velocities consistently lower than those of the 1000-hPa observational results of Heymsfield and Kajikawa (1987) for a given diameter; Heymsfield and Kajikawa (1987) observations may encompass a spectrum of “stellar” shapes, whereas the calculated values in this study are from idealized shapes.

Fig. 11.
Fig. 11.

Terminal velocities (cm s−1) of planar crystals for 900 hPa. Square, circle, diamond, and triangle markers indicate the results for sector plates (P1b), broad-branched crystals (P1c), stellar crystals (P1d), and ordinary dendritic crystals (P1e), respectively. The corresponding thick curves are the power-law fits given by Eqs. (8)(11): solid curve for sector plates, dashed curve for broad-branched crystals, dotted curve for sector plates, and dash–dotted curve for dendrites. Thick, gray solid curve represents the 1000-hPa parameterization of hexagonal plates (P1a) from Cheng et al. (2015). The corresponding 1000-hPa terminal velocity parameterizations of Heymsfield and Kajikawa (1987) are shown by thin curves, with colors and line styles matching these experimental results.

Citation: Journal of the Atmospheric Sciences 75, 9; 10.1175/JAS-D-18-0041.1

The calculated 900-hPa terminal velocities of each crystal type can be fit by the following power-law relationships with diameter:
e8
e9
e10
e11
where has units of meters per second and d is in micrometers. The root-mean-square error of the fits are 0.020, 0.017, 0.008, and 0.017 m s−1, respectively. The relationships are valid over the range of diameters studied, 2005000 μm for 900 hPa. Relationships between the terminal velocity and crystal surface area are as follows:
e12
e13
e14
e15
where has units of meters per second and sa is the crystal surface area in square meters. The root-mean-square error of the fits are 0.060, 0.047, 0.008, and 0.017 m s−1, respectively. The relationships are valid over the range of diameters studied, 2005000 μm for 900 hPa.

e. Drag coefficients

Figure 12 shows the time-averaged drag coefficient CD as a function of Reynolds number, where the drag has been computed as the average CD over the last 3000 time steps, as described above. Drag coefficient data from other studies are presented in Fig. 12 for comparison, including numerical results for hexagonal plates (P1a) from Cheng et al. (2015), Hashino et al. (2014), and Wang and Ji (1997) and for broad-branched crystals (P1c), also from Wang and Ji (1997).

Fig. 12.
Fig. 12.

Drag coefficient of planar crystals. Square, circle, diamond, and triangle markers indicate the results for sector plates (P1b), broad-branched crystals (P1c), stellar crystal (P1d), and ordinary dendritic crystals (P1e), respectively. The corresponding curves are the power-law fits given by Eqs. (16)(19): solid curve for sector plates, dashed curve for broad-branched crystals, dotted curve for sector plates, and dash–dotted curve for dendrites. Bolded bullets represent the steady-state drag coefficients for hexagonal plates (P1a) from Hashino et al. (2014). Thick gray solid curve represents the parameterization of hexagonal plates from Cheng et al. (2015). The parameterizations of Wang and Ji (1997) are shown in thin lines: solid curve for hexagonal plates and dashed curve for broad-branched crystals. The adjusted Cd values discussed in Heymsfield and Westbrook (2010) are shown as the filled-in markers, colors and shapes corresponding to the crystal type described in the legend.

Citation: Journal of the Atmospheric Sciences 75, 9; 10.1175/JAS-D-18-0041.1

The drag coefficient for the crystal types in this study can be fit by the following two-term power-law relationships:
e16
e17
e18
e19
where all variables are dimensionless. The root-mean-square error of the fits are 0.404, 0.193, 179.9, and 1.884, respectively, with adjusted r2 values of 0.9924, 0.9996, 0.9686, and 0.9991, respectively. Power-law relationships were selected because polynomial fits might overfit the data and are not representative of a realistic relationship. The relationships are valid over the range of Re corresponding to each crystal type, found in Table 1.
Table 1.

Dimensions, dimensionless moments of inertia, Reynolds numbers, terminal velocities, and ventilation coefficients for the planar ice crystals investigated in this study.

Table 1.

The drag coefficients from the studies of Böhm (1989) and Heymsfield and Westbrook (2010) are also shown in Fig. 12. All their Cd values are lower than that of the idealized crystals studied here. The discrepancy is most likely due to the difference in idealized versus observed crystal shapes.

Figure 13 shows the relation between the Reynold number and the Best number X for the crystals studied here as well as those by Böhm (1989) and Heymsfield and Westbrook (2010). The Best number is defined as
e20
where NScυ is the Schmidt number of water vapor (NScυ = air kinematic viscosity/water vapor diffusivity) The Best number is useful in the heat and mass transfer to and from the crystal surface, as we will discuss in the next section. The discrepancy between the present and previous studies of Böhm (1989) and Heymsfield and Westbrook (2010) is again most likely due to shape differences.
Fig. 13.
Fig. 13.

Relation between the Reynolds number and Best number for the crystals studied here as well as those by Böhm (1989) and Heymsfield and Westbrook (2010).

Citation: Journal of the Atmospheric Sciences 75, 9; 10.1175/JAS-D-18-0041.1

f. Vapor density distribution

The ventilation effect, the enhancement of the diffusional growth/evaporation rate of cloud and precipitation particles due to movement relative to the flow, is due to the enhancement of the vapor density gradient around a falling hydrometeor compared to a stationary one (Hall and Pruppacher 1976; Ji and Wang 1999; Pruppacher and Klett 1997; Wang 2013).

The computed vapor density distributions in the yz plane around the 5-mm-diameter crystals studied are shown in Fig. 14. In common with all simulations, areas sufficiently far from the crystals and the upstream region are characterized by high vapor density equal to the environmental 2% supersaturation condition, with lower vapor density in the downstream region. This is caused by the motion of the ice crystals; a stationary crystal in such an environment would have a symmetric vapor density distribution. In the downstream wake, there is an asymmetry in the vapor distribution for the sector plate (P1b) and broad-branched crystal (P1c) (Re = 384 and 385, respectively) and general symmetry for the stellar (P1d) and ordinary dendritic crystals (P1e) (Re = 133 and 173, respectively). The stronger return flows of the sector plate and broad-branched crystal act to transport the water vapor closer to the surface of the crystal in the wake, leading to a tighter gradient downstream. Important for the diffusional growth of an ice crystal is the gradient of the vapor density, with the largest gradient corresponding to the highest local diffusion rate (Cheng et al. 2014). The highest vapor density gradients are seen upstream of the crystal, though the gradient downstream is certainly not negligible. Note that the patterns shown change with time for unsteady flow, though the patterns and characteristics described above remain similar over time.

Fig. 14.
Fig. 14.

Vapor density distributions (g m−3) around planar crystals with maximum dimension (diameter) of 5 mm. Vapor density is shown by shades of blue (darker: approaching 2% supersaturation ρυ = 2.4086 g m−3; lighter: approaching saturation ρυ = 2.3613 g m−3). All snapshots correspond to the same randomly selected time step, as in Fig. 2.

Citation: Journal of the Atmospheric Sciences 75, 9; 10.1175/JAS-D-18-0041.1

The results of the mean ventilation coefficient are summarized in Fig. 15 and include numerical results from Ji and Wang (1999) and Hall and Pruppacher (1976). The results from the current study can be fit by the following empirical formulas, taking a form similar to that of Ji and Wang (1999):
e21
e22
e23
e24
where is the dimensionless ventilation coefficient.
Fig. 15.
Fig. 15.

Ventilation coefficient of planar crystals. Square, circle, diamond, and triangle markers indicate the results for sector plates (P1b), broad-branched crystals (P1c), stellar crystals (P1d), and ordinary dendritic crystals (P1e), respectively. The corresponding curves are the power-law fits given by Eqs. (20)(23): solid curve for sector plates, dashed curve for broad-branched crystals, dotted curve for sector plates, and dash–dotted curve for dendrites. The parameterizations of Ji and Wang (1999) are shown in thin lines: dotted curve for circular columns, solid curve for hexagonal plates, and dashed curve for broad-branched crystals. The general expression for the ventilation coefficient based on data for spheres and oblate spheroids from Hall and Pruppacher (1976) is shown in the thin black dash–dotted line.

Citation: Journal of the Atmospheric Sciences 75, 9; 10.1175/JAS-D-18-0041.1

The functional dependence of on X has been found by Pitter et al. (1974). As in Ji and Wang (1999), is held at a constant value (=0.63 for the atmospheric conditions considered here), so the relationships in Eqs. (21)(24) are essentially between and Re. The root-mean-square errors for the above relationships are 0.034, 0.018, 0.014, and 0.028, respectively, and are valid over the range of Re corresponding to each crystal type, found in Table 1.

Figure 15 also shows that the ordinary dendritic crystals (P1e) generally have the higher ventilation coefficient, at a given Reynolds number, compared to the other crystal habits. This becomes more pronounced with increasing Re. This can be understood by considering the dimensions and structure of the varying crystal habits. The more skeletal structure of a dendrite allows for a greater surface area that can be subjected to the ventilation effect despite falling at a lower terminal velocity compared to the sector plate (P1b) and broad-branched crystal (P1c) at the same Re.

4. Summary and outlook

The numerical simulations of the hydrodynamic behavior of four types of freely falling planar ice crystals are performed in this study. The crystals range from 0.2 to 5 mm in maximum diameter and cover both steady and unsteady flow regimes. Fall behavior, flow characteristics, and an analysis of the results are reported and are in general agreement with previous numerical studies and reported observations. Allowing the crystals to respond to the forcing of the flow field, as first done in Cheng et al. (2015) for a hexagonal plate, provides for more realistic results and works to improve the understanding of frozen precipitation particles in clouds. These results can be parameterized for use by cloud and numerical weather prediction models. More realistic ventilation representation should improve the accuracy of numerical cloud models.

Only idealized, symmetric crystals are simulated. Observations show that natural ice crystals do not demonstrate near-perfect symmetry most of the time and often have rough surfaces. Simulations can easily be run for different environmental conditions (temperature and pressure) for comparison. Additionally, individual ice crystals often form aggregates with other crystals while they fall toward the surface of Earth. Simulating the flow fields around crystal aggregates is the next phase of this study. This endeavor will prove more challenging to mesh and simulate, though early trials from a colleague in the laboratory are promising.

Acknowledgments

This study is partially supported by the U.S. National Science Foundation (NSF) Grant AGS-1633921 and research fund provided by the Academia Sinica, Taiwan. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.

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