Simple Analytical Expressions for Steady-State Vapor Growth and Collision–Coalescence Particle Size Distribution Parameter Profiles

Edwin L. Dunnavan; Alexander V. Ryzhkov

doi:10.1175/JAS-D-23-0052.1

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Journal of the Atmospheric Sciences

Abstract
- Significance Statement
1. Introduction
2. The general steady-state moment equation
3. Reflectivity-based solutions
- a. Collision–coalescence only
- b. Vapor deposition only
4. General height solution for vapor deposition and collision–coalescence
5. Comparison with observations
- a. Lagrangian spiral descent aircraft profiles
- b. Parsivel disdrometer measurements with quasi-vertical polarimetric radar profiles
6. Discussion and conclusions
Acknowledgments.
Data availability statement.
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Fig. 1.

Contour plots of (top) $F_{β_{m}} (μ, b_{υ})$ and (bottom) $G_{β_{m}} (μ, b_{υ})$ . Columns correspond to (left) β_m = 2.0, (center) β_m = 2.5, and (right) β_m = 3.0.

View raw image

Fig. 2.

As in Fig. 1, but for $H_{β_{m}} (μ, b_{υ})$ .

View raw image

Fig. 3.

Contour plots of (top) Eq. (41) and (bottom) Eq. (44) (scaled by 10⁶) for different values of μ, b_υ, β_m. Columns correspond to (left) β_m = 2.0, (center) β_m = 2.5, and (right) β_m = 3.0. Thick black lines in the bottom row plots correspond to C_tot = 0.

View raw image

Fig. 4.

Scatterplots of logN_t (N_t in units of L⁻¹) vs logD_m (D_m in units of mm) for spiral 2 (blue) and spiral 3 (orange). Linear fits to each scatterplot are shown as solid lines with slope −C(μ, b_υ, β_m) where C(μ, b_υ, β_m) = 3.204 (spiral 2) and C(μ, b_υ, β_m) = 2.945 (spiral 3). The intercept of each linear fit is adjusted to best fit each scatterplot data.

View raw image

Fig. 5.

(top) Comparison of CRYSTAL-FACE Lagrangian spiral aircraft data for spiral 2 (black) and spiral 3 (gray) with corresponding steady-state expressions using Eq. (43). Solid lines for the observations correspond to mean profiles derived by excessively smoothing the spiral data. Steady-state model lines correspond to the 50 sets of parameters with the lowest N_t and D_m errors between the model and measurements when optimizing $\bar{E}$ . Steady-state N_t results are calculated using the same size range, D = 0.1–27 mm, as the aircraft measurements. (bottom) Boxplots of the 50 sets of model parameters used in the steady-state model for each spiral. Blue lines/boxplots correspond to spiral 2 whereas orange lines/boxplots correspond to spiral 3.

View raw image

Fig. 6.

(from top to bottom) Time series of KASPR QVP radar reflectivity, KASPR QVP specific differential phase, and Parsivel disdrometer measurement of N_t and D_m on 1 Feb 2021. Heights are above mean sea level. KASPR radar data were linearly interpolated to the Parsivel’s 1-min-time-resolution grid. Both Parsivel and KASPR measurements were smoothed with a 2-h moving-average filter. The steady-state model N_t was calculated in the same interval as provided by the Parsivel disdrometer: 0.2 ≤ D ≤ 25 mm. Minimum and maximum Parsivel measurements were determined through the use of an assumed Poisson uncertainty in particle detection as described in McFarquhar et al. (2015).

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Editorial Type:: Article

Article Type:: Research Article

Simple Analytical Expressions for Steady-State Vapor Growth and Collision–Coalescence Particle Size Distribution Parameter Profiles

Edwin L. Dunnavan

Edwin L. Dunnavan ^aCooperative Institute for Severe and High-Impact Weather Research and Operations, Norman, Oklahoma
^bNOAA/OAR/National Severe Storms Laboratory, Norman, Oklahoma

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https://orcid.org/0000-0002-2730-216X

and

Alexander V. Ryzhkov

Alexander V. Ryzhkov ^aCooperative Institute for Severe and High-Impact Weather Research and Operations, Norman, Oklahoma
^bNOAA/OAR/National Severe Storms Laboratory, Norman, Oklahoma

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Online Publication:: 12 Oct 2023

Print Publication:: 01 Oct 2023

DOI:: https://doi.org/10.1175/JAS-D-23-0052.1

Page(s):: 2531–2544

Received:: 24 Mar 2023

Final Form:: 26 Jul 2023

Accepted:: 27 Jul 2023

Published Online:: 12 Oct 2023

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

This study derives simple analytical expressions for the theoretical height profiles of particle number concentrations (N_t) and mean volume diameters (D_m) during the steady-state balance of vapor growth and collision–coalescence with sedimentation. These equations are general for both rain and snow gamma size distributions with size-dependent power-law functions that dictate particle fall speeds and masses. For collision–coalescence only, N_t (D_m) decreases (increases) as an exponential function of the radar reflectivity difference between two height layers. For vapor deposition only, D_m increases as a generalized power law of this reflectivity difference. Simultaneous vapor deposition and collision–coalescence under steady-state conditions with conservation of number, mass, and reflectivity fluxes lead to a coupled set of first-order, nonlinear ordinary differential equations for N_t and D_m. The solutions to these coupled equations are generalized power-law functions of height z for D_m(z) and N_t(z) whereby each variable is related to one another with an exponent that is independent of collision–coalescence efficiency. Compared to observed profiles derived from descending in situ aircraft Lagrangian spiral profiles from the CRYSTAL-FACE field campaign, these analytical solutions can on average capture the height profiles of N_t and D_m within 8% and 4% of observations, respectively. Steady-state model projections of radar retrievals aloft are shown to produce the correct rapid enhancement of surface snowfall compared to the lowest-available radar retrievals from 500 m MSL. Future studies can utilize these equations alongside radar measurements to estimate N_t and D_m below radar tilt elevations and to estimate uncertain microphysical parameters such as collision–coalescence efficiencies.

Significance Statement

While complex numerical models are often used to describe weather phenomenon, sometimes simple equations can instead provide equally good or comparable results. Thus, these simple equations can be used in place of more complicated models in certain situations and this replacement can allow for computationally efficient and elegant solutions. This study derives such simple equations in terms of exponential and power-law mathematical functions that describe how the average size and total number of snow or rain particles change at different atmospheric height levels due to growth from the vapor phase and aggregation (the sticking together) of these particles balanced with their fallout from clouds. We catalog these mathematical equations for different assumptions of particle characteristics and we then test these equations using spirally descending aircraft observations and ground-based measurements. Overall, we show that these mathematical equations, despite their simplicity, are capable of accurately describing the magnitude and shape of observed height and time series profiles of particle sizes and numbers. These equations can be used by researchers and forecasters along with radar measurements to improve the understanding of precipitation and the estimation of its properties.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Edwin L. Dunnavan, edwin.dunnavan@noaa.gov

Abstract

Significance Statement

Corresponding author: Edwin L. Dunnavan, edwin.dunnavan@noaa.gov

Keywords: Atmosphere; Cloud microphysics; In situ atmospheric observations; Differential equations; Numerical analysis/modeling; Single column models

1. Introduction

Analytical methods and models have been widely used to simplify and explain complex atmospheric equations. Even in recent years, there have been a number of studies that have developed and reported on simple expressions that can be derived from the equations of motion (e.g., Du and Rotunno 2014; Davies-Jones 2022) as well as from the kinetic equation for the growth of precipitation particles (e.g., Garrett 2019; Wu and McFarquhar 2018). The appeal of such simplified, analytic formulas stems from multiple factors. For example, analytical models can provide efficient results compared to computationally expensive numerical weather prediction models and provide benchmark formulas to validate these numerical models. Simple analytic models and theoretical frameworks can also be invaluable for providing insight and understanding of physical phenomena itself through explicit mathematical relationships among various variables.

One such example of an analytical model is the so-called snow growth model developed by Passarelli (1978a,b), Mitchell (1988, 1991), and Mitchell et al. (2006). The snow growth models developed in these papers represent height profiles for particle size distribution (PSD) parameters under steady-state snowfall conditions. PSDs are assumed a priori with number distribution functions n(D) within the gamma family

n (D) = N_{0} D^{μ} e^{- Λ D},

(1)

where D is the equivolume diameter, N₀ is the intercept parameter, and Λ is the slope parameter. PSD moments are given by

M_{r} \equiv \int_{D = 0}^{\infty} D^{r} n (D) d D,

(2)

and thus can be used to express important bulk quantities such as number concentration (i.e., N_t ≡ M₀) and mean volume diameter D_m ≡ M₄/M₃. Passarelli (1978a,b) and Mitchell (1988) assumed that μ = 0, thus specifying an exponential distribution whereas Mitchell (1991) and Naakka (2015) used a variable μ. These studies then assume a steady-state condition [i.e.,

\partial n (D, t) / \partial t \approx 0

], negligible horizontal advection and small vertical wind speeds compared to the particle fall speed (i.e., |w| ≪ υ_t). With these assumptions, the change in the number distribution function due to vapor deposition and collision–coalescence is

\begin{array}{l} \frac{\partial [υ_{t} n (x, z)]}{\partial z} = - \frac{\partial}{\partial x} [\frac{d x}{d t} n (x, z)] \\ + \frac{1}{2} \int_{y = 0}^{x} n (x - y, z) n (y, z) K (x - y, y) d y - n (x, z) \int_{y = 0}^{\infty} n (y, z) K (x, y) d y, \end{array}

(3)

where n(x, z) is the mass distribution function [a power-law transformation of n(D, z)], υ_t is the particle fall speed defined as positive downward, x and y are particle masses for two interacting particles, z is height which is below some assumed reference top z_top, and K(x, y) is an integration kernel that dictates the physics of particle-to-particle collision and coalescence. The left-hand side of Eq. (3) represents vertical advection, the first term on the right-hand side is vapor deposition, and the last two terms represent the gain and loss terms of collision–coalescence.

Notice that integrating both sides of Eq. (3) using the kernel xⁿ, where n is an arbitrary moment order, develops expressions for the steady-state vertical moment fluxes

\begin{matrix} \frac{d}{d z} [\int_{x = 0}^{\infty} υ_{t} x^{n} n (x, z) d x] = - \int_{x = 0}^{\infty} x^{n} \frac{\partial}{\partial x} [\frac{d x}{d t} n (x, z)] d x \\ + \frac{1}{2} \int_{x = 0}^{\infty} \int_{y = 0}^{\infty} [{(x + y)}^{n} - x^{n} - y^{n}] \times K (x, y) n (x, z) n (y, z) d x d y, \end{matrix}

(4)

where the collision–coalescence gain and loss terms are combined in the bracketed seconded term on the right-hand side in Eq. (4) by interchanging the inner and outer integrals, performing a substitution of variables and by exploiting the symmetry in x and y [see Thompson (1968) and Pruppacher and Klett (1997) for more details]. With some algebraic manipulations, it is possible to use Eq. (4) to develop equations for dM_n/dz which can then be used to develop expressions for Λ(z) and N₀(z). Older studies such as Passarelli (1978a) and Mitchell (1988, 1991) have neglected using the number concentration flux (i.e., n = 0) in Eq. (4) and instead have developed equations for the PSD slope parameter Λ(z) and N_t(z) by evolving mass (i.e., n = 1) and reflectivity (i.e., n = 2) related moments. More recent studies have incorporated n = 0 in their snow growth models and further introduced additional terms such as nucleation (Mitchell et al. 2006; Naakka 2015). By conserving the total mass and reflectivity in Eq. (4), these studies showed that the resulting solutions for Λ(z) can be expressed in terms of solutions to Bernoulli’s differential equation. While these are closed-form solutions, they can be quite involved to compute and are often numerically integrated depending upon the complexity of an assumed vapor mass flux term (e.g., Passarelli 1978a; Mitchell 1988; Naakka 2015).

After nearly half a century of research, there are still a number of unanswered questions regarding how these steady-state models can potentially aid in understanding precipitation processes as well as their practical utility. Previous studies regarding the development and use of snow growth models (Lo and Passarelli 1982; Mitchell 1988; Mitchell et al. 2006) have noted that plots of logN₀ versus logΛ are roughly linear during regions of pronounced aggregation from in situ measurements. These studies have qualitatively described this behavior yet none of these studies have shown quantitatively why this behavior manifests. Furthermore, there is still much work to be done regarding the use of such steady-state models. Perhaps the most promising utility of such simple models is the potential to use radar measurements to project microphysical retrievals into regions that are below what radars are able to remotely sense. For example, the snow growth model of Mitchell et al. (2006) was able to show good agreement between measured snow precipitation rates from surface gauges and from a snow growth model that was initialized from radar reflectivity aloft.

This study simplifies the snow growth model approach and its associated equations as much as possible such that its use is more accessible to a wider audience. Such a simplification also promotes a better physical understanding of such steady-state conditions since the solutions we derive are in the form of simple exponential and generalized power-law functions. As we will show, using the number-, mass-, and reflectivity-related moments in Eq. (4) allows for N_t and D_m (which are directly related to Λ and N₀) to be solved for a number of various assumptions and approximations. Moreover, these solutions are functions of height z and depend only upon four factors: the combined collision–coalescence efficiency $\bar{E}$ , μ, and the mass– and fall speed–size power-law exponents β_m and b_υ which appear in the equations: $m (D) = α_{m} D^{β_{m}}$ and $υ_{t} (D) = a_{υ} D^{b_{υ}}$ . While these four parameters depend upon particle characteristics which can depend upon height, we assume that these four quantities can be replaced by representative mean quantities that are constant for all heights. Steady-state solutions can be expressed not only in terms of z, but also in terms of reflectivity Z for the special cases of vapor growth only or collision–coalescence only. Unlike height-based solutions of N_t and D_m, reflectivity-based solutions do not depend at all on $\bar{E}$ . Height-based solutions, when evaluated against descending Lagrangian spiral in situ probe data (section 5), appear to be quite robust and capable of producing the correct shape and magnitude of the corresponding height profiles for appropriate parameter values associated with ice–ice aggregates. The following section derives a simplified form of Eq. (4) that dictates the vertical gradients of each PSD moment.

2. The general steady-state moment equation

The vapor deposition term in Eq. (4) can be integrated by parts where the bracketed term is the differential (see Passarelli 1978a). Doing so gives

\begin{array}{l} \frac{d}{d z} [\int_{x = 0}^{\infty} υ_{t} x^{n} n (x, z) d x] = n \int_{x = 0}^{\infty} x^{n - 1} \frac{d x}{d t} n (x, z) d x \\ + \frac{1}{2} \int_{x = 0}^{\infty} \int_{y = 0}^{\infty} [{(x + y)}^{n} - x^{n} - y^{n}] \times K (x, y) n (x, z) n (y, z) d x d y . \end{array}

(5)

Notice that Eq. (5) introduces a factor of n that results from taking the derivative of xⁿ. We can directly replace instances of masses x and y with equivolume diameters D_x and D_y using

m (D) = α_{m} D^{β_{m}}

in Eq. (5). Notice that α_m cancels from both sides if α_m is assumed constant for all heights. The m–D relation also implies that mass moments correspond to size moments as r = β_mn, where r refers to moment orders of the size distribution. As a result, we can rewrite Eq. (5) as

\begin{array}{l} \frac{d}{d z} [\int_{D_{x} = 0}^{\infty} υ_{t} (D_{x}) D_{x}^{r} n (D_{x}, z) d D_{x}] = \frac{r}{β_{m}} \frac{f (z)}{α_{m}} M_{r + 1 - β_{m}} + \frac{1}{2} \int_{D_{x} = 0}^{\infty} \int_{D_{y} = 0}^{\infty} [{(D_{x}^{β_{m}} + D_{y}^{β_{m}})}^{r / β_{m}} - D_{x}^{r} - D_{y}^{r}] \times K (D_{x}, D_{y}) n (D_{x}, z) n (D_{y}, z) d D_{x} d D_{y}, \end{array}

(6)

where we assume that

d x / d t \approx f (z) D

and f(z) is an environmental factor that depends upon supersaturation and diffusion. In situ aircraft studies such as Field et al. (2006) indicate that this assumption is reasonable for steady-state vapor deposition of snowflakes in tropical anvil clouds.

As done by Mitchell (1988) and others, we can use the mean value theorem of integration to separate the fall speed component from moment height gradients, dM_r/dz, on the left-hand side of Eq. (6). This produces

\begin{matrix} ⟨ υ_{t, r} ⟩ \frac{d M_{r}}{d z} = \frac{r}{β_{m}} \frac{f (z)}{α_{m}} M_{r + 1 - β_{m}} \\ + \frac{1}{2} \int_{D_{x} = 0}^{\infty} \int_{D_{y} = 0}^{\infty} [{(D_{x}^{β_{m}} + D_{y}^{β_{m}})}^{r / β_{m}} - D_{x}^{r} - D_{y}^{r}] \times K (D_{x}, D_{y}) n (D_{x}, z) n (D_{y}, z) d D_{x} d D_{y} \end{matrix},

(7)

where

⟨ υ_{t, r} ⟩ = \frac{\int_{D_{x} = 0}^{\infty} υ_{t} (D_{x}) D_{x}^{r} n (D_{x}, z) d D_{x}}{\int_{D_{x} = 0}^{\infty} D_{x}^{r} (D_{x}, z) d D_{x}}

(8)

are moment-weighted mean fall speeds. At this point, Eq. (7) is general for any PSD. However, we can constrain an assumed PSD for n(D, z) such that the parameters take on the z dependence. This is essentially how Passarelli (1978a), Lo and Passarelli (1982), Mitchell (1988), and others developed expressions for the height evolution of Λ and N₀.

In this study, we also assume a gamma distribution that is parameterized as

n (D, z) = \frac{N_{t} (z)}{Γ [μ (z) + 1]} \frac{1}{D_{m} (z)} {[μ (z) + 4]}^{1 + μ (z)} {[\frac{D}{D_{m} (z)}]}^{μ (z)} \times \exp {- \frac{[μ (z) + 4] D}{D_{m} (z)}},

(9)

where

M_{r} (z) = N_{t} (z) {[μ (z) + 4]}^{- r} D_{m}^{r} (z) \frac{Γ [μ (z) + 1 + r]}{Γ [μ (z) + 1]} .

(10)

Using this gamma distribution form, we can represent 〈υ_t_,_r〉 as

⟨ υ_{t, r} ⟩ (z) = a_{υ} (z) {[μ (z) + 4]}^{- b_{υ} (z)} D_{m}^{b_{υ} (z)} \frac{Γ [μ (z) + b_{υ} (z) + 1 + r]}{Γ [μ (z) + 1 + r]} .

(11)

Putting Eq. (11) in Eq. (7), using the hydrodynamic kernel

K (D_{x}, D_{y}) = (π / 4) {(D_{x} + D_{y})}^{2} | υ_{t} (D_{x}) - υ_{t} (D_{y}) | = a_{υ} (π / 4) {(D_{x} + D_{y})}^{2} | D_{x}^{b_{υ}} - D_{y}^{b_{υ}} |

, and rearranging to solve for dM_r/dz gives

\begin{array}{l} \frac{d M_{r}}{d z} = \frac{r}{β_{m}} \frac{f (z)}{α_{m} a_{υ}} M_{r + 1 - β_{m}} {(μ + 4)}^{b_{υ}} D_{m}^{- b_{υ}} \frac{Γ (μ + 1 + r)}{Γ (μ + 1 + r + b_{υ})} \\ + \frac{π}{8} N_{t}^{2} {(μ + 4)}^{- (r + 2)} D_{m}^{r + 2} \bar{E} \frac{Γ (μ + 1 + r)}{Γ^{2} (μ + 1) Γ (μ + 1 + r + b_{υ})} \times I_{r} (μ, b_{υ}, β_{m}) ({mm}^{r + 2} m^{- 6}), \end{array}

(12)

where I_r is

I_{r} (μ, b_{υ}, β_{m}) \equiv \int_{x = 0}^{\infty} \int_{y = 0}^{\infty} x^{μ} y^{μ} [{(x^{β_{m}} + y^{β_{m}})}^{r / β_{m}} - x^{r} - y^{r}] \times {(x + y)}^{2} | x^{b_{υ}} - y^{b_{υ}} | e^{- x} e^{- y} d x d y .

(13)

Here, Eq. (12) is simplified by assuming

μ (z) \approx \bar{μ} = μ

and

b_{υ} (z) \approx {\bar{b}}_{υ} = b_{υ}

. Equation (13) can be expressed in closed form in terms of a finite series of Gauss’s hypergeometric function ₂F₁(a, b; c; z) if r/β_m is an integer and in terms of an infinite series of Gauss’s hypergeometric functions if r/β_m is fractional or irrational (see Tables 1–3 for a list of all equations and parameters used throughout). For example, I₀ and

I_{2 β_{m}}

can be expressed in closed form as

I_{0} = - J_{0} (μ, b_{υ}),

(14a)

I_{2 β_{m}} = 2 J_{β_{m}} (μ, b_{υ}),

(14b)

where

\begin{matrix} J_{n} = 2 \sum_{k = 0}^{2} \sum_{l = 0}^{1} (\begin{matrix} 2 \\ k \end{matrix}) {(- 1)}^{l} \frac{Γ [2 (μ + n) + b_{υ} + 4]}{b_{υ} l - b_{υ} - μ - n - k - 1} \\ \times {}_{2}F_{1} [μ + n + k + b_{υ} - b_{υ} l + 1, 2 (μ + n) + b_{υ} + 4; μ + n + k + b_{υ} - b_{υ} l + 2; - 1] . \end{matrix}

(15)

While it is often convenient to numerically integrate Eq. (13) itself, many modern programming languages have standard libraries that can calculate Gauss’s hypergeometric function. For example, Gauss’s hypergeometric function is available in the Python scipy.special library as “hyp2f1” whereas MATLAB can evaluate it using the generalized “hypergeom” function.

Table 1.

Equations used throughout this study. Units are as follows: z, meters; N_t_,top, per cubic meter; D_m_,top, millimeters.

Table 2.

Simplified equations that can be derived from Table 1 for snow: μ = 0, β_m = 2.0, and b_υ = 0.2; and rain: μ = 0, β_m = 3.0, and b_υ = 0.65. Units are as follows: N_t, per liter; z, kilometers.

Table 3.

Parameters and constants used in this study.

Equation (12) further simplifies by considering

d M_{β_{m}} / d z

, which is proportional to the vertical mass flux. Collision–coalescence does not change

M_{β_{m}}

and all changes come from vapor deposition. Therefore, this gives

\begin{array}{l} \frac{d M_{β_{m}}}{d z} = \frac{f (z)}{α_{m} a_{υ}} {(μ + 4)}^{b_{υ}} D_{m}^{- b_{υ}} M_{1} \frac{Γ (μ + β_{m} + 1)}{Γ (μ + b_{υ} + β_{m} + 1)} . \end{array}

(16)

It is now possible to rearrange Eq. (16) in order to replace unknown terms, like f(z), with

d M_{β_{m}} / d z

\frac{f (z)}{α_{m} a_{υ}} {(μ + 4)}^{b_{υ}} D_{m}^{- b_{υ}} = \frac{d M_{β_{m}}}{d z} \frac{1}{M_{1}} \frac{Γ (μ + b_{υ} + β_{m} + 1)}{Γ (μ + β_{m} + 1)} .

(17)

Equation (12) can now be put into a final form

\begin{matrix} \frac{d M_{r}}{d z} = \frac{r}{β_{m}} \frac{d M_{β_{m}}}{d z} \frac{M_{r + 1 - β_{m}}}{M_{1}} \frac{Γ (μ + b_{υ} + β_{m} + 1)}{Γ (μ + β_{m} + 1)} \\ \times \frac{Γ (μ + 1 + r)}{Γ (μ + b_{υ} + 1 + r)} + 10^{- 6} \frac{π}{8} M_{r} M_{2} \bar{E} \\ \times \frac{I_{r} (μ, b_{υ}, β_{m})}{Γ (μ + 3) Γ (μ + b_{υ} + 1 + r)} ({mm}^{r} m^{- 4}), \end{matrix}

(18)

where the factor of 10⁻⁶ in the collision–coalescence term is needed for consistent units of mm^r m⁻⁴ and where we have used Eq. (10) to put everything in terms of gamma distribution moments. The following sections utilize Eq. (18) to develop steady-state expressions for N_t and D_m under various conditions. It is worth noting that Eq. (18) is general for any moment order r, and thus there is ambiguity as to which moments are used to solve for D_m and N_t. For this study, we use the number (r = 0) and reflectivity-related moments (r = 2β_m) as constraints for deriving steady-state solutions for N_t and D_m. Note that the mass moment r = β_m is implicitly included as a constraint through the use of Eqs. (16) and (17).

3. Reflectivity-based solutions

For particles comparatively smaller than a radar’s wavelength λ (i.e., Rayleigh scattering regime), radar reflectivity (in units of mm⁶ m⁻³) can be expressed as (cf. Ryzhkov et al. 1998)

Z_{h} = g (α_{m}, λ) M_{2 β_{m}},

(19)

where g(α_m, λ) is a constant that depends upon the dielectric properties of the particle which are influenced by both α_m and λ. Note that for raindrops, g(α_m, λ) does not directly depend upon α_m. In decibels, this is

Z (dB Z) \equiv 10 \log_{10} (Z_{h}) = \frac{10}{\ln 10} \ln Z_{h} .

(20)

The following sections use Eqs. (18)–(20) to develop simple expressions for N_t(Z) and D_m(Z).

a. Collision–coalescence only

Considering only collision–coalescence by ignoring the vapor growth term in Eq. (18) and using r = 0 and 2β_m leads to the following equations:

\frac{d M_{0}}{d z} = \frac{d N_{t}}{d z} = 10^{- 6} \frac{π}{8} N_{t} M_{2} \bar{E} \frac{I_{0} (μ, b_{υ})}{Γ (μ + 3) Γ (μ + b_{υ} + 1)},

(21a)

\frac{d M_{2 β_{m}}}{d z} = 10^{- 6} \frac{π}{8} M_{2 β_{m}} M_{2} \bar{E} \frac{I_{2 β_{m}} (μ, b_{υ}, β_{m})}{Γ (μ + 3) Γ (μ + b_{υ} + 2 β_{m} + 1)} .

(21b)

Dividing Eq. (21b) by

M_{2 β_{m}}

reduces Eq. (21b) to the same order as Eq. (21a). This leads to

\begin{array}{l} \frac{1}{M_{2 β_{m}}} \frac{d M_{2 β_{m}}}{d z} = 10^{- 6} \frac{π}{8} M_{2} \bar{E} \frac{I_{2 β_{m}} (μ, b_{υ}, β_{m})}{Γ (μ + 3) Γ (μ + b_{υ} + 2 β_{m} + 1)} . \end{array}

(22)

We can use Eq. (20) and the property of logs in Eq. (22) to put Eq. (22) in a revealing form in terms of Z:

\frac{1}{M_{2 β_{m}}} \frac{d M_{2 β_{m}}}{d z} \equiv \frac{1}{Z_{h}} \frac{d Z_{h}}{d z} = \frac{d \ln Z_{h}}{d z} = \frac{\ln (10)}{10} \frac{d [10 \log_{10} (Z_{h})]}{d z} \equiv \frac{\ln (10)}{10} \frac{d Z}{d z} .

(23)

Now we can divide Eq. (21a) by Eq. (22) in order to eliminate D_m and

\bar{E}

entirely:

\frac{d N_{t}}{d Z} = \frac{\ln (10)}{10} N_{t} \frac{Γ (μ + b_{υ} + 2 β_{m} + 1)}{Γ (μ + b_{υ} + 1)} \frac{I_{0} (μ, b_{υ})}{I_{2 β_{m}} (μ, b_{υ})} .

(24)

We can solve Eq. (24) using an initial condition at some height reference (i.e., a retrieved or assumed N_t at z_top). The solution to Eq. (24) is an exponential given by

N_{t} (Z) = N_{t, top} e^{- F_{β_{m}} (μ, b_{υ}) (Z - Z_{top})},

(25)

where

F_{β_{m}} (μ, b_{υ}) = \frac{\ln 10}{20} \frac{Γ (μ + b_{υ} + 2 β_{m} + 1)}{Γ (μ + b_{υ} + 1)} \frac{J_{0} (μ, b_{υ})}{J_{β_{m}} (μ, b_{υ})} .

(26)

We can develop a similar expression for D_m(Z) by using the definition of

M_{2 β_{m}}

from Eq. (10), which can be used to derive the equation

Z_{h} = Z_{h, top} {(D_{m} / D_{m, top})}^{2 β_{m}} N_{t} / N_{t, top}

. This produces

D_{m} (Z) = D_{m, top} e^{G_{β_{m}} (μ, b_{υ}) (Z - Z_{top})},

(27)

where

G_{β_{m}} (μ, b_{υ}) = \frac{\ln (10)}{20 β_{m}} [1 + \frac{1}{2} \frac{Γ (μ + b_{υ} + 2 β_{m} + 1)}{Γ (μ + b_{υ} + 1)} \frac{J_{0} (μ, b_{υ})}{J_{β_{m}} (μ, b_{υ})}] .

(28)

Bulk parameters R such as IWC or S can be similarly put into the same form as Eqs. (25) and (27):

R (Z) = R_{top} e^{[r G_{β_{m}} (μ, b_{υ}) - F_{β_{m}} (μ, b_{υ})] (Z - Z_{top})} .

(29)

Figure 1 shows the behavior of

F_{β_{m}} (μ, b_{υ})

and

G_{β_{m}} (μ, b_{υ})

for different values of μ, b_υ, and β_m. μ values are shown from −0.8 to 5.0 because N_t is specifically only valid from D = 0 to D = ∞ for μ > −1 and the calculation of I₀ and

I_{2 β_{m}}

become difficult to numerically evaluate for μ < −0.8 and μ > 5. This range represents many gamma fit observations including Brandes et al. (2007). For simplicity, Fig. 1 shows only β_m = 2 (corresponding to unrimed snowflakes), β_m = 2.5 (corresponding to rimed, hexagonal, or quasi-spherical ice particles), and β_m = 3.0 (corresponding to rain or dense, quasi-spherical ice particles).

Fig. 1.

Contour plots of (top) $F_{β_{m}} (μ, b_{υ})$ and (bottom) $G_{β_{m}} (μ, b_{υ})$ . Columns correspond to (left) β_m = 2.0, (center) β_m = 2.5, and (right) β_m = 3.0.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0052.1

For both

F_{β_{m}} (μ, b_{υ})

and

G_{β_{m}} (μ, b_{υ})

, the largest values are given for the broadest distributions (i.e., lower μ values) with the largest b_υ and β_m values.

F_{β_{m}} (μ, b_{υ})

is more sensitive to each parameter and has larger minimum and maximum values than

G_{β_{m}} (μ, b_{υ})

. This behavior results from N_t being equally sensitive to all sizes. In particular, the lower end of the size distribution when μ < 0 can dramatically influence N_t. D_m is instead weighted by the largest particles which is dictated predominantly by the exponential component of the assumed gamma PSD. Moreover,

G_{β_{m}} (μ, b_{υ})

is not as sensitive to β_m when b_υ is low but quite sensitive to β_m when b_υ is high. For unrimed aggregate snowflakes which are known to have β_m ≈ 2.0 that are roughly in the range 1.9 ≤ β_m ≤ 2.2 (Mitchell 1996; Schmitt and Heymsfield 2010) and low values of b_υ (Mitchell and Heymsfield 2005; Brandes et al. 2008), the values of F₂(μ, b_υ) and G₂(μ, b_υ) do not vary much for different μ values. For unrimed snowflake aggregates with μ = 0 and b_υ = 0.2 (cf. Brandes et al. 2008), these equations simplify to

N_{t} \approx N_{t, top} e^{- 0.23 (Z - Z_{top})},

(30a)

D_{m} \approx D_{m, top} e^{0.11 (Z - Z_{top})} .

(30b)

It is interesting to note that similar equations can be derived from the consideration of precipitation (mass flux) conservation and gamma distribution continuity throughout the total layer. This can be done by combining the expression for snow rate or snowfall mass flux [see Eq. (10.66) from Ryzhkov and Zrnić 2019] and Eqs. (19) and (20). Using μ = 0, b_υ = 0.2, and β_m = 2.0 with this snow rate conservation from Ryzhkov and Zrnić (2019) produces

N_{t} (Z) = N_{t, top} e^{- 0.28 (Z - Z_{top})},

(31)

which is quite similar to Eq. (30a).

b. Vapor deposition only

For vapor deposition only,

d N_{t} / d z = 0

so N_t(z) = N_t_,top. The same general strategy as in the collision–coalescence only case can be used to derive D_m(Z). Assuming

\bar{E} = 0

in Eq. (18), and using r = 2β_m, we can divide through by

M_{2 β_{m}}

to yield

\frac{d \ln M_{2 β_{m}}}{d z} = \frac{2}{M_{2 β_{m}}} \frac{d M_{β_{m}}}{d z} \times \frac{M_{β_{m + 1}}}{M_{1}} \frac{Γ (μ + b_{υ} + β_{m} + 1) Γ (μ + 2 β_{m} + 1)}{Γ (μ + β_{m} + 1) Γ (μ + b_{υ} + 2 β_{m} + 1)} .

(32)

Using the definition of

M_{2 β_{m}}

from Eq. (10), Eq. (20) and rearranging for dD_m/dz gives

\frac{d D_{m}}{d z} D_{m}^{β_{m} - 1} = \frac{\ln 10}{20 β_{m}} \frac{Γ (μ + 2) Γ (μ + b_{υ} + 2 β_{m} + 1)}{Γ (μ + β_{m} + 2) Γ (μ + b_{υ} + β_{m} + 1)} \frac{d Z}{d z} .

(33)

Integrating Eq. (33) from D_m_,top to D_m yields

D_{m} = {[D_{m, top}^{β_{m}} + H_{β_{m}} (μ, b_{υ}) (Z - Z_{top})]}^{1 / β_{m}},

(34)

where

H_{β_{m}} (μ, b_{υ}) = \frac{\ln 10}{20} \frac{Γ (μ + 2) Γ (μ + b_{υ} + 2 β_{m} + 1)}{Γ (μ + β_{m} + 2) Γ (μ + b_{υ} + β_{m} + 1)} .

(35)

Figure 2 shows values of

H_{β_{m}} (μ, b_{υ})

. As in Fig. 1, low b_υ values that describe snow aggregates appear to show relatively low variability in H₂(μ, b_υ). However, there is more variability in H₃(μ, b_υ) for changes in both μ and b_υ.

Fig. 2.

As in Fig. 1, but for $H_{β_{m}} (μ, b_{υ})$ .

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0052.1

For unrimed snowflakes (β_m = 2) with μ = 0 and b_υ = 0.2, Eq. (34) reduces to

D_{m} \approx \sqrt{D_{m, top}^{2} + 0.26 (Z - Z_{top})},

(36)

whereas for rain (β_m = 3) with μ = 0 and b_υ = 0.65, Eq. (34) reduces to

D_{m} \approx \sqrt[3]{D_{m, top}^{3} + 0.84 (Z - Z_{top})} .

(37)

4. General height solution for vapor deposition and collision–coalescence

For both vapor deposition and collision–coalescence, dM₀/dz and

d M_{2 β_{m}} / d z

can close the set of equations to solve for N_t(z) and D_m(z). Substituting M_r[N_t(z), D_m(z)] in Eq. (18) using Eq. (10), utilizing the product rule for the derivative terms, and rearranging each equation produces the following coupled nonlinear differential equation set:

\frac{d D_{m}}{d z} = C_{D_{m}} \bar{E} N_{t} D_{m}^{3},

(38a)

\frac{d N_{t}}{d z} = - C_{N_{t}} \bar{E} N_{t}^{2} D_{m}^{2},

(38b)

where

\begin{array}{l} C_{D_{m}} = 10^{- 6} \frac{π}{16 β_{m}} {(μ + 4)}^{- 2} \frac{1}{Γ (μ + 1)} \times \frac{\frac{2 J_{β_{m}} (μ, b_{υ})}{Γ (μ + b_{υ} + β_{m} + 1)} - [2 \frac{Γ (μ + β_{m} + 2)}{Γ (μ + 2)} - \frac{Γ (μ + b_{υ} + 2 β_{m} + 1)}{Γ (μ + b_{υ} + β_{m} + 1)}] \frac{J_{0} (μ, b_{υ})}{Γ (μ + b_{υ} + 1)}}{\frac{Γ (μ + b_{υ} + 2 β_{m} + 1)}{Γ (μ + b_{υ} + β_{m} + 1)} - \frac{Γ (μ + β_{m} + 2)}{Γ (μ + 2)}} \end{array},

(39a)

C_{N_{t}} = 10^{- 6} \frac{π}{8} {(μ + 4)}^{- 2} \frac{J_{0} (μ, b_{υ})}{Γ (μ + 1) Γ (μ + b_{υ} + 1)} .

(39b)

Dividing Eq. (38b) by Eq. (38a) produces

\frac{d \ln N_{t}}{d \ln D_{m}} = - \frac{C_{N_{t}}}{C_{D_{m}}} \equiv - C (μ, b_{υ}, β_{m}),

(40)

where

C (μ, b_{υ}, β_{m}) = 2 β_{m} \frac{\frac{Γ (μ + b_{υ} + 2 β_{m} + 1)}{Γ (μ + b_{υ} + β_{m} + 1)} - \frac{Γ (μ + β_{m} + 2)}{Γ (μ + 2)}}{2 \frac{Γ (μ + β_{m} + 2)}{Γ (μ + 2)} - \frac{Γ (μ + b_{υ} + 2 β_{m} + 1)}{Γ (μ + b_{υ} + β_{m} + 1)} - 2 \frac{Γ (μ + b_{υ} + 1)}{Γ (μ + b_{υ} + β_{m} + 1)} \frac{J_{β_{m}} (μ, b_{υ})}{J_{0} (μ, b_{υ})}} .

(41)

Integrating Eq. (40) leads to the result

N_{t} (z) = N_{t, top} {[\frac{D_{m} (z)}{D_{m, top}}]}^{- C (μ, b_{υ}, β_{m})} .

(42)

Now we can use Eq. (42) in Eq. (38) in order to solve for D_m(z) and N_t(z):

D_{m} (z) = D_{m, top} {[1 + C_{tot} \bar{E} N_{t, top} D_{m, top}^{2} (z_{top} - z)]}^{C_{D_{m}} / C_{tot}},

(43a)

N_{t} (z) = \frac{N_{t, top}}{{[1 + C_{tot} \bar{E} N_{t, top} D_{m, top}^{2} (z_{top} - z)]}^{C_{N_{t}} / C_{tot}}},

(43b)

where

C_{tot} = C_{N_{t}} - 2 C_{D_{m}} .

(44)

Notice that any bulk parameter R can be expressed with Eqs. (10) and (43) as

R (z) = R_{top} {[1 + C_{tot} \bar{E} N_{t, top} D_{m, top}^{2} (z_{top} - z)]}^{C_{R} / C_{tot}},

(45)

where

C_{R} = r C_{D_{m}} - C_{N_{t}} .

(46)

In this context, R could be PSD moments such as N_t or microphysical quantities such as precipitation rates. For example, Eq. (45) can be used to express precipitation rate S and ice water content (IWC) as

S (z) = S_{top} {[1 + C_{tot} \bar{E} N_{t, top} D_{m, top}^{2} (z_{top} - z)]}^{C_{S} / C_{tot}},

(47a)

IWC (z) = {IWC}_{top} {[1 + C_{tot} \bar{E} N_{t, top} D_{m, top}^{2} (z_{top} - z)]}^{C_{IWC} / C_{tot}},

(47b)

where

C_{S} = (β_{m} + b_{υ}) C_{D_{m}} - C_{N_{t}}

and

C_{IWC} = β_{m} C_{D_{m}} - C_{N_{t}}

Figure 3 shows the behavior of Eqs. (41) and (44) for different values of β_m, μ, and b_υ. As in Figs. 1 and 2, C(μ, b_υ, β_m) is largest when β_m = 3 and smallest when β_m = 2. C(μ, b_υ, 3) varies from approximately 0.71 to 4.16 when β_m = 2 and from approximately 2.44 to 6.0 when β_m = 3. This range is even smaller if μ = 0 where C(μ, b_υ, 2) varies between approximately 1.62 and 2.85 and C(μ, b_υ, 3) varies between approximately 4.22 and 5.28. In general, for a given β_m value, C(μ, b_υ, β_m) is approximately constant for linear changes in μ and b_υ. For lower (more negative) values of μ, the slope of constant C(μ, b_υ, β_m) values becomes steeper.

Fig. 3.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0052.1

There is also a fascinating feature of Eq. (43) that is worth mentioning. Both equations depend upon the quantity C_tot, which appears in both the bracketed term and the power-law exponent. Taking the limit of each equation as C_tot → 0 produces the elegant equations

D_{m} (z) = D_{m, top} e^{C_{D_{m}} \bar{E} N_{t, top} D_{m, top}^{2} (z_{top} - z)},

(48a)

N_{t} (z) = N_{t, top} e^{- C_{N_{t}} \bar{E} N_{t, top} D_{m, top}^{2} (z_{top} - z)} .

(48b)

Figure 3 shows the contour lines for C_tot = 0 which is apparent only for β_m = 2.0 and 2.5 within the shown μ and b_υ ranges. For β_m = 2.0, this line appears to quite accurately represent aggregates where observations have frequently shown −0.05 ≤ b_υ ≤ 0.2 (Locatelli and Hobbs 1974; Mitchell and Heymsfield 2005; Brandes et al. 2008) and low (even negative) values of μ (Brandes et al. 2007; Duffy and Posselt 2022). For denser ice particles where β_m = 2.5 (cf. Mitchell 1996), the C_tot = 0 line exists for −0.05 ≤ b_υ ≤ 0.6 and 2 ≤ μ ≤ 5. Equation (48) could therefore possibly be used to describe ice particle populations with these higher values of μ for certain environments.

For easy reference, Table 1 presents all derived equations in this paper, Table 2 presents all simplified versions of the derived equations, and Table 3 presents all associated constants and parameters.

5. Comparison with observations

a. Lagrangian spiral descent aircraft profiles

Two spiral descent aircraft legs from the 26 July 2002 CRYSTAL-FACE field campaign case were used to evaluate the Eq. (43) solutions for snow particles. This particular dataset was used by Field et al. (2006) alongside a spectral bin microphysical model to study the effects of self-collection of ice particles and more details about the data can be found in that study. For the current study, we only use data from the second and third Lagrangian spiral descents, which were within stratiform conditions.

Figure 4 displays scatterplots of log(N_t) versus log(D_m) from the Lagrangian spiral descent measurements for both spirals 2 and 3. Each set of measurements is fitted with a line where the slope theoretically corresponds to −C(μ, b_υ, β_m) as suggested by Eq. (40). While there is an inconsistency in that the measured N_t is calculated from a truncated size range (i.e., D = 0.1 to 27 mm), both spirals do show a roughly linear relationship between logN_t and logD_m. This linear relationship is most apparent for spiral 2 where r² = 0.9244 whereas r² = 0.6558 for spiral 3.

Fig. 4.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0052.1

To evaluate N_t and D_m profiles predicted by Eq. (43), the C(μ, b_υ, β_m) values from Fig. 4 were used as constraints for choosing μ, b_υ, and β_m values for the steady-state model profiles shown in Fig. 5. To do this, 3D surfaces of μ, b_υ, and β_m were constructed for each C(μ, b_υ, β_m) value shown in Fig. 4. Five hundred random points on these surfaces were then used as the chosen parameters for the steady-state model for each spiral. For each of these 500 points, $\bar{E}$ was optimized for Eq. (43) by performing a nonlinear least squares fitting by using the N_t(z) and D_m(z) mean observed profiles (shown as solid black and gray lines in Fig. 5) as observational constraints. All model profiles used N_t_,top and D_m_,top from these mean observational profiles.

Fig. 5.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0052.1

Figure 5 shows 50 of the 500 N_t(z) and D_m profiles with the lowest errors compared to the mean observed profiles. Figure 5 also shows boxplots of the associated steady-state model parameters for all 50 profiles. Both the profiles and boxplots indicate an overall higher sensitivity of these solutions for spiral 2 than for spiral 3. The reason for this is due to the higher values of N_t during spiral 2. This is because the aggregation process scales as $N_{t}^{2}$ as shown in Eq. (23). Therefore, the small changes in $\bar{E}$ for spiral 2 cause larger differences in N_t(z) and D_m(z) compared to spiral 3. Both steady-state model results show qualitatively consistent N_t(z) and D_m(z) behavior compared to the mean observed profiles where N_t(z) decreases and D_m(z) increases with descending z approximately as generalized power laws. While the overall shape of N_t(z) profiles are consistent for both spiral 2 and 3, the shapes of the D_m(z) profiles are only consistent for spiral 3 where D_m increases almost linearly with height. Spiral 2, on the other hand, has a slight upward concavity, which is not captured by the steady-state model. The optimal parameter sets between each spiral are rather different as shown in the Fig. 5 boxplots. μ is strictly negative for spiral 2 with values less than −0.5 whereas spiral 3 has a larger spread of values with a median value of 0. β_m ≈ 1.9 for spiral 2 is characteristic of the well-known Brown and Francis (1995) mass–dimensional power-law exponent characteristic of cirrus clouds. Spiral 3, on the other hand, has larger β_m values more typical of rimed particles. For spiral 2, the 50 theoretical N_t and D_m profiles have mean relative errors of 7.85% and 3.79%, respectively. Mean relative errors for spiral 3 are somewhat lower where N_t and D_m profiles have errors of 6.85% and 1.33%.

b. Parsivel disdrometer measurements with quasi-vertical polarimetric radar profiles

The Stony Brook University KASPR research radar was used to generate quasi-vertical profiles (QVPs; Ryzhkov et al. 2016) for the 1 February 2021 snowstorm. QVPs are azimuthally averaged radar measurements that are plotted in a time–height format. KASPR QVPs were calculated for reflectivity Z and specific differential phase K_dp and the simplified steady-state height profiles from Table 2 were calculated using the Z/K_dp Dunnavan et al. (2022, their Table 1) retrievals of D_m_,top and N_t_,top assuming α = 0.2 g cm⁻³ mm, σ = 20°, and $\bar{φ} = 0.6$ . Figure 6 shows the QVP time series for Z and K_dp as well as the corresponding Parsivel disdrometer measurements of N_t and D_m at the surface. Notice that Z increases and K_dp decreases from 3.0 km toward ground which is indicative of a snow aggregation signature (Schrom et al. 2015; Dunnavan et al. 2022). Also, due to the radar blind range, the QVP observations are only available for heights above about 500 m MSL. Therefore, while the QVP observations cannot be used for the N_t and D_m retrieval near the surface, the steady-state model can be used instead. Because dual-polarization retrievals are most accurate when polarimetric signatures are prominent (Dunnavan et al. 2022), we initialize the steady-state model at z_top = 3.0 km where K_dp reaches its highest values. We also plot retrievals of each variable using the Z and K_dp time series from the 500 m level.

Fig. 6.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0052.1

Figure 6 surface measurements of N_t and D_m show a slow, steady increase in the presence of snow throughout the day from 0200 to 1400 UTC. This can be best seen in N_t which slowly increases during this period. Snowfall rapidly increases from 1400 to 1800 UTC which can be seen from all four time series plots. The simplified, exponential steady-state model from Table 2 based on Eq. (48) with $\bar{E} = 0.14$ was used along with the 3.0 km MSL Z/K_dp retrievals to predict the associated PSD moment evolution at the surface. Unlike 500 m retrievals which show a near constant N_t at the surface and a gradual increase in D_m, the steady-state model shows the correct evolution of snow properties from 1400 to 1800 UTC. The primary reason for this good agreement stems from the prominent K_dp signature at 3.0 km which leads to a better-predicted increase in N_t starting around 1400 UTC. This increase in N_t leads to associated increases in D_m primarily through aggregation. Such a prominent K_dp increase is not present at 500 m because snow aggregates tend to mute the K_dp signature.

6. Discussion and conclusions

The equations derived in this paper as shown in Tables 1–3 can be used to easily estimate PSD and bulk parameters such as precipitation rates in regions where radar retrievals are either difficult to perform due to instrumental uncertainties or in regions where radar variables are unavailable. One such application example would be the use of the height-based solutions of N_t(z) and D_m(z) for regions below the lowest radar tilt available from WSR-88Ds where there are no available data to infer precipitation properties. Figure 6 shows that N_t and D_m at the surface can be estimated from QVP profiles initialized aloft and, with appropriate parameters, can produce representative profiles of each microphysical variable. Remarkably, the simplified, exponential version of Eq. (43) in Table 2 is capable of accurately capturing the rapid onset of surface snowfall and the associated increases in microphysical variables. This rapid increase in surface variables is not captured by the lowest level radar retrievals due to the muted and variable K_dp signature.

The equations shown in Table 1 and throughout this paper also explain common observations of the relationship between N_t and D_m from previous studies. For example, Figs. 5 and 7 from Lo and Passarelli (1982) show that N₀/Λ phase-space trajectories of snow from Lagrangian spiral data during the aggregation stage are quasi linear in log–log space. Lo and Passarelli (1982) explain in their study that the quasi-linear behavior (in log–log space) of N₀ and Λ indicate constant moments M_r of an exponential distribution (i.e., μ = 0) if the slope equals r + 1. However, Lo and Passarelli (1982) do not explain the quantitative or theoretical reasons as to why aggregation produces the quasi-linear behavior. Since N_t = N₀Λ⁻⁽^μ⁺¹⁾ and D_m = (μ + 4)/Λ for gamma PSDs, the steady-state model [Eq. (42)] phase space for N_t and D_m would be exactly linear in log–log space where −C(μ, b_υ, β_m) corresponds to the observed slope (Fig. 4). As a result, it may be possible to determine approximate steady-state conditions from Lagrangian spiral descent data through linear fits of logN_t and logD_m or logN₀ and logΛ as indicated in plots like Fig. 4. These linear fits could then be used to catalog the associated C(μ, b_υ, β_m) values and gamma distribution fits to the in situ PSDs can be used to constrain μ. Indeed, as suggested by the scatterplot measurements of logD_m versus logN_t, observations do indicate a linear relationship with a constant slope. This is most apparent for spiral 2 where data points are close to the linear fit (r² = 0.9244) but also for spiral 3, albeit with a worse overall fit (r² = 0.6558).

The steady-state equations in this work can also be used to estimate

\bar{E}

. For example, Passarelli (1978b) was able to estimate

\bar{E}

from his steady-state equations by rearranging his solution for Λ(z) and using in situ aircraft cloud probe data to close the equation for

\bar{E}

. Similarly, there are a number of possible ways to use the equations in Table 1 to solve for

\bar{E}

. One possible estimation method for

\bar{E}

would be to utilize Eq. (48) and dual-polarization radar retrievals for N_t and D_m (e.g., Ryzhkov and Zrnić 2019; Dunnavan et al. 2022). For example, using Z and specific differential phase K_dp for either N_t or D_m relation from Table 1 of Dunnavan et al. (2022) allows for a simple

\bar{E}

retrieval for steady-state snowfall within a predetermined height layer

{\bar{E} |}_{z_{bottom}}^{z_{top}} = \frac{\ln (\frac{Y_{top}}{Y_{bottom}})}{C_{Y} N_{t, top} D_{m, top}^{2} (z_{top} - z_{bottom})},

(49)

where Y is either N_t or D_m. Dual-polarization radar retrieval equations such as from Ryzhkov and Zrnić (2019) or Dunnavan et al. (2022) can be used to determine N_t_,top and D_m_,top.

While the equations derived in this work are promising, they rely upon a number of important assumptions and limitations. First, it is assumed that $\bar{E}$ , μ, b_υ, and β_m are constant throughout a particular layer. $\bar{E}$ depends upon both environmental conditions such as temperature and relative humidity as well as particle characteristics such as type and size (Phillips et al. 2015). μ, b_υ, and β_m similarly can spatiotemporally vary depending upon meteorological condition. Even within the same air volume, different snow particle types can coexist and the proposed steady-state analytical model in this work does not account for such diversity. Furthermore, the use of r = 0 (i.e., the use of dM₀/dz = dN_t/dz) in the formulation of the equations in Table 1 is subject to potentially large errors stemming from uncertainties in n(D) for small particle sizes. These uncertainties are exacerbated by negative values of μ, which lead to sharp increases in n(D) for D → 0. The above equations also neglect a number of important microphysical processes such as riming, nucleation, and breakup which can also impact N_t(z) and D_m(z). It is possible that, despite these important uncertainties and limitations, the equations presented in this work can describe, on average, the appropriate behavior of N_t and D_m. Future work can explore and report on optimized parameter sets for different meteorological scenarios.

Acknowledgments.

The authors thank Mariko Oue (Stony Brook University) for the KASPR and Parsivel disdrometer data used in this study and Petar Bukovčić for his comments and suggestions. Funding was provided by NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA21OAR4320204, U.S. Department of Commerce, and by National Science Foundation Grant AGS-1841246.

Data availability statement.

Aircraft data from the CRYSTAL-FACE campaign are available from https://data.eol.ucar.edu/project/127. The KASPR and Parsivel disdrometer data are not publicly available and must be requested directly from the Stony Brook University Radar Science Group.

REFERENCES

Brandes, E. A., K. Ikeda, G. Zhang, M. Schönhuber, and R. M. Rasmussen, 2007: A statistical and physical description of hydrometeor distributions in Colorado snowstorms using a video disdrometer. J. Appl. Meteor. Climatol., 46, 634–650, https://doi.org/10.1175/JAM2489.1.
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Journal of the Atmospheric Sciences

Abstract
- Significance Statement
1. Introduction
2. The general steady-state moment equation
3. Reflectivity-based solutions
- a. Collision–coalescence only
- b. Vapor deposition only
4. General height solution for vapor deposition and collision–coalescence
5. Comparison with observations
- a. Lagrangian spiral descent aircraft profiles
- b. Parsivel disdrometer measurements with quasi-vertical polarimetric radar profiles
6. Discussion and conclusions
Acknowledgments.
Data availability statement.
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Fig. 1.

Contour plots of (top) $F_{β_{m}} (μ, b_{υ})$ and (bottom) $G_{β_{m}} (μ, b_{υ})$ . Columns correspond to (left) β_m = 2.0, (center) β_m = 2.5, and (right) β_m = 3.0.