Table 1
Nomenclature.
| SYMBOL | MEANING |
|---|---|
| A | Crystal area (m2) |
| a | Hexagonal column crystal radius (m) |
| Bi | Total radiance of black body (W m–2) |
| c | Hexagonal column crystal half-length (m) |
| cB | Hexagonal column crystal core half-length (m) |
| cpa | Specific heat of dry air (J K–1 kg–1) |
| C | Capacitance of the ice crystal (m) |
| CD | Drag coefficient |
| Cp | Phase speed of oscillations (m s–1) |
| Di | Equivalent diameter of ice particle (m) |
| Dv | Diffusivity of water vapour in air (m2 s–1) |
| fv | Ventilation coefficient |
| Fu, Fd | Upward, downward radiative flux densitities (W m–2) |
| g | Gravity field constant (g = 9.81 m s–2) |
| Hl | Relative humidity of water vapour over liquid water |
| Hi | Relative humidity of water vapour over ice |
 | Heating rate (K d–1) |
| IWP | Ice Water Path (g m–3) |
| k | Damping coefficient (s–1) |
| Ka | Heat conductivity of ambient dry air (W K–1 m–1) |
| Ls | Latent heat of sublimation of ice (Ls = 2.837 × 106 J kg–1) |
| Lv | Latent heat of vaporization of water (Lv = 2.525 × 106 J kg–1) |
| m | Crystal mass (kg) |
| P | Period of oscillatory motion (s) |
| p | Total pressure (Pa) |
| pr | Reference pressure (pr = 105 Pa) |
| pa | Partial pressure of dry air (Pa) |
| pv | Partial pressure of water vapour (Pa) |
| pvsl | Saturation pressure of water vapour over liquid water (Pa) |
| pvsi | Saturation pressure of water vapour over ice (Pa) |
| qs | Saturation mixing ratio (kg kg–1) |
| Qabs | Efficiency factor for absorption of radiation (Qabs ≈ 1) |
| ri | Equivalent radius of ice crystal (m) |
| Source term in radiative contribution (W) | |
| R | Radiative term in the mass equation (%) |
| Ra | Gas constant of dry air (Ra = 287.05 J K–1 kg–1) |
| Rv | Gas constant of water vapour (Rv = 461.00 J K1 kg–1) |
| Re | Reynolds number |
| Ri | Richardson number |
| Sl | Saturation ratio of water vapour over liquid water |
| Si | Saturation ratio of water vapour over ice |
| T | Absolute temperature (K) |
| Tr | Reference temperature (Tr = 273.15 K) |
| Ta | Temperature of ambient air (K) |
| Ts | Temperature of crystal surface (K) |
| u, w | Components of crystal velocity (m s–1) |
| Ua, Wa | Components of ambient air velocity (m s–1) |
| V | Crystal volume (m3) |
| Wf | Free-fall speed (m s–1) |
| Wf∞ | Terminal free-fall speed (m s–1) |
| x, z | Cartesian coordinates of ice crystal (m) |
| za0 | Critical altitude of horizontal wind (m) |
| zw0 | Critical altitude of updraft (m) |
| α | Coefficient in Eq. (27) |
| ε | Ratio of gas constants Ra/Rv (ε = 0.622) |
| ζ | Damping ratio |
| η | Radiative transfer ratio |
| θ | Latitude |
| θd, θe | Potential temperature (dry, equivalent) |
| Λ | Wavelength of oscillatory motion (m) |
| μa | Dynamic viscosity of air (Pa s) |
| ρa | Mass density of dry air (kg m–3) |
| ρ | Mass density of wet air (kg m–3) |
| ρi | Mass density of ice (ρi = 920 kg m–3) |
| σ | Stefan-Boltzmann constant (σ = 5.67 × 10–8 W m–2 K–4) |
| σS | Vertical gradient of ice supersaturation (m–1) |
| τ | Non-dimensional time of analytic model |
| τD | Drag relaxation time (s) |
| ϕ | Aspect ratio (ϕ = c/a) |
| ψ | Hollowness factor (ψ = 1–cB/c) |
| ω | Wind shear (m s–1 km–1 or s–1) |
| Ω | Angular frequency of oscillations (= 2π/P) (rad s–1) |
| Ω0 | Rotation rate of the Earth (Ω0= 7.27 × 10–5 rad s–1) |
| ΩBV | Brunt-Väisälä angular frequency (rad s–1) |
Figure 1
Atmospheric profile of horizontal wind.
Figure 2
Atmospheric profiles of vertical wind.
Figure 3
Atmospheric profiles of temperature and pressure of dry air.
Figure 4
Atmospheric profiles of relative humidity and saturation ratios.
Figure 5
Atmospheric profile of atmospheric infrared ratio ηa.
Figure 6
The shape of the mean crystal (meridian and axial cross sections) (after Chen and Wang, 2009).
Table 2
Parameters of test-cases/numerical experiments.
| CASE NB | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| NET RAD. TRANSF. | R ≠ 0 | R = 0 | R ≠ 0 | R ≠ 0 |
| Parameter | η(z) ϕ = ϕ0 cB = cB0 | η = η0 = 1 ϕ = ϕ0 cB = cB0 | η = η0 ≠ 1 ϕ = ϕ0 cB = cB0 | η = η0 ≠ 1 ϕ = ϕ0 cB = cB0 |
| Modes | Steady – Osc | Steady – Osc | (Steady) – Osc | (Steady) – Osc |
| Life time | 5 h/40 h | 5 h/40 h | 4 h/40 h | 4 h/40 h |
| ∆t (s) | 0.02/0.04 | 0.02/0.04 | 0.02/0.04 | 0.02/0.04 |
| x0 (km) | 2 | 2 | 2 | 2 |
| z0 (km) | 10 | 10 | 10 | 10 |
| u0 (m s–1) | 0 | 0 | 0 | 0 |
| w0 (m s–1) | 0.6 | 0.6 | 0.8 | 0.8 |
| a0 (μm) | 51.1 | 50.17 | 52.0 | 50.2 |
| c0 (μm) | 102.2 | 100.34 | 104.0 | 100.4 |
| ϕ0 | 2.0 | 2.0 | 2.0 | 2.0 |
| ψ0 | 0.80 | 0.80 | 0.80 | 0.80 |
| cB0 (μm) | 20.44 | 20.068 | 20.80 | 20.08 |
| m0 (μg) | 0.935 | 0.885 | 0.986 | 0.887 |
| C0 (μm) | 88.6 | 86.9 | 90.1 | 87.0 |
| D0 (μm) | 125 | 122 | 127 | 123 |
| ρie0 (kg m–3) | 675 | 675 | 675 | |
| η | 0.9 ≤ η ≤ 1.1 | 1 | 1.1 | 0.9 |
| zη0 (km) | 9 ≤ z ≤ 10 | 0 – 11 | 0 – 11 | 0 – 11 |
| Wa0 (m/s) | 0.6 | 0.6 | 0.8 | 0.8 |
| zW0 (km) | 10/9 | 10/9 | 10/7 | 10/7 |
| Ha3 | 0.61 | 0.61 | 0.68 | 0.68 |
| zH3 (km) | 10 | 10 | 10 | 10 |
| zmax0–z0 (m) | 102 | 144 | 468 | 542 |
| z∞(km) | 9.63 | 9.58 | 8.58 | 8.3 |
| P (hour) | 10.2 | 8.87 | 7.8 | 6.8 |
| Λ (km) | 69.8 | 64.8 | 200 | 200 |
| Cp (m s–1) | 1.85 | 2.03 | 7.1 | 8.2 |
| ζ | 0.019 | 0.015 | 0.021 | 0.011 |
Figure 7
Profiles of the parcel trajectory and hodograph (z0 = 10 km; zw0 = 10 km).
Figure 8
Time profiles of abscissa and altitude (zw0 = 10 km).
Figure 9
Time profiles of free-fall speed Wf and driving factor Si–1–R (zw0 = 10 km).
Figure 10
Profiles of ice supersaturation Si–1 and radiative correction R (zw0 = 10 km).
Figure 11
Profiles of crystal’s mass and dimensions (zw0 = 10 km).
Figure 12
Profiles of mass equivalent diameter and dynamic viscosity (zw0 = 10 km).
Figure 13
Profiles of hollowness factor and effective mass density of crystal (zw0 = 10 km).
Figure 14
Profiles of drag coefficient and Reynolds number (zw0 = 10 km).
Figure 15
Time profiles of damping coefficient and temperature difference ice-air (zw0 = 10 km).
Figure 16
Profiles of the parcel trajectory and hodograph (z0 = 10 km; zw0 = 9 km).
Figure 17
Time profiles of abscissa and altitude (zw0 = 9 km).
Figure 18
Time profiles of free-fall speed Wf and driving factor Si–1–R (zw0 = 9 km).
Figure 19
Time profiles of ice supersaturation Si-1 and radiative correction R (zw0 = 9 km).
Figure 20
Time profiles of mass m and dimensions a, c, cB (zw0 = 9 km).
Figure 21
Time profiles of mass equivalent diameter and dynamic viscosity (zw0 = 9 km).
Figure 22
Time profiles of hollowness factor and effective mass density of crystal (zw0 = 9 km).
Figure 23
Time profiles of drag coefficient CD and Reynolds number Re (zw0 = 9 km).
Figure 24
Time profiles of damping coefficient k and temperature difference ice-ambient air (zw0 = 9 km).
Figure 25
Profiles of falling height z20–z0 and maximum height in the head zM–z0 as functions of crystal half-width a0 (zw0 = 10 km).
Figure 26
Profiles of transit times t10, t20 and half-lengths c10/c0 and c20/c0 as functions of crystal half-width a0 (zw0 = 10 km).
Figure 27
Profiles of pseudo-periods between minima (Pmin) and maxima (Pmax) estimated with Eq. (53), as functions of crystal half-width a0 (zw0 = 9 km).
Figure 28
Profiles of wavelengths Λmin and Λmax derived from distances of minima and maxima by Eq. (54) as functions of crystal half-width a0 (zw0 = 9 km).
Figure 29
Profiles of phase speeds Cpmin and Cpmax derived by Eq. (55) as functions of crystal half-width a0 (zw0 = 9 km).
Figure 30
Profiles of relative altitudes at minima and maxima in the trail as functions of crystal half-width a0 (zw0 = 9 km).
Figure 31
Profiles of relative limit height z∞–z0 and velocity Ua(z∞) in the trail as functions of crystal size a0 (zw0 = 9 km).
Figure 32
Profiles of the parcel trajectory, hodograph, altitude, supersaturation (η = 1; Wa = 0.6 m/s; Ha3 = 61%; zw0 = 9 km).
Figure 33
Profiles of the parcel trajectory, hodograph, altitude, supersaturation Si –1, driving factor Si –1–R and radiative correction R (η = 1.1; Wa = 0.8 m/s; Ha3 = 68%; zw0 = 7 km).
Figure 34
Time profile of the horizontal velocity u (η = 1.1; Wa = 0.8 m/s; Ha3 = 68%; zw0 = 7 km).
Figure 35
Profiles of the parcel trajectory, hodograph, altitude, supersaturation Si –1, driving factor Si –1–R and radiative correction R (η = 0.9; Wa = 0.8 m/s; Ha3 = 68%; zw0 = 7 km).
Figure 36
Time profile of the horizontal velocity u (η = 0.9; Wa = 0.8 m/s; Ha3 = 68%; zw0 = 7 km).
Figure 37
Atmospheric profiles of Richardson numbers.
Figure 38
Qualitative time profile of analytical free fall speed.
Figure 39
Trajectory and hodograph for analytic solution, Eq.(99).
Figure 40
Time profile of free fall speed for analytic solution, Eq.(99).
Figure 41
Altitude time profile of theoretical underdamped harmonic oscillator, Eq.(118) for moderate (a) and strong (b) damping.