Reference solutions to complex weather phenomena provide a useful way to develop and evaluate new methods for retrieving microphysical variables from weather radar measurements. Such retrievals can involve simple analytic formulas e.g., or more involved machine learning models e.g., that connect radar products like reflectivity to microphysical parameters such as rain or snow rates. Typically, these retrieval equations are developed through a combination of theory and observations, but often without rigorous consideration of microphysical and scattering uncertainties as well as the precise nature of the simulated particle size distribution (PSD). Instead, parameteric approaches, often by assuming an underlying exponential or gamma distribution, are used to connect observed radar variables to desirable microphysical quantities. As a result, however, these parametric approaches sever the important connections between the quantities that are inferred from radar signatures with their underlying microphysical causes.

Of the many microphysical processes in precipitation physics, collision-coalescence and collisional breakup represent two of the most important yet difficult to explicitly simulate. This difficulty can be seen directly from the general equation for collision-coalesence and collsional breakup in time and in height (ignoring horizontal advection terms), which can be expressed as cf. 1∂n(m,z,t)∂t+∂vt(m)n(m,z,t)dz=12∫x=0mC(m-x,x)n(m-x,z,t)n(x,z,t)dx-n(m,z,t)∫x=0∞C(m,x)n(x,z,t)dx+12∫x=0∞∫y=0∞B(x,y)P(m|x,y)n(x,z,t)n(y,z,t)dydx-n(m,z,t)∫y=0∞B(m,y)n(y,z,t)m+ydy∫x=0m+yxP(x|m,y)dx, where n(m,z,t) is the number distribution function for mass m, height z, and time t; vt(m) is the fall speed for particles of mass m; C(x,y) and B(x,y) are the collision-coalescence and collision-breakup kernels, respectively; and P(m|x,y) is the conditional fragment distribution that determines the number of mass m fragments generated from collisions involving particles of mass x and y. The left hand side of Eq. () represents the tendency and sedimentation terms of the evolving PSD (in terms of mass) whereas the first two right hand side terms represent the gain and loss terms for coalescence and the last two terms represent the gain and loss terms for breakup. Equation () reduces to the well-known Smoluchowski equation when the sedimentation and breakup terms are neglected. The Smoluchowski equation is also frequently referred to by many other names such as the coagulation equation, the stochastic collection equation , the kinetic collection equation and the scalar transport equation . Equation () is difficult to solve for several reasons. First, the integro-differential equation is functional in the coalescence gain term through C(m-x,x) and n(m-x,z,t). As such, standard finite difference numerical methods are not sufficient for directly solving Eq. (). Secondly, the integral regions are constrained by the particle masses themselves as seen in Eq. () with the coalescence gain term and the breakup loss term. This entails that different integration methods are potentially required for different particle m masses. Finally, there are large uncertainties involving the nature and form of the breakup conditional distribution P(m|x,y). By its nature, P(m|x,y) should respect detailed mass conservation in the loss term (i.e., ∫x=0m+yxP(x|m,y)=m+y), however this condition isn't always exactly met . For rain collisional breakup, this has led to several intricate methods that use mixture distributions involving, for example, Gaussian and lognormal distributions to represent the different types of breakup events (i.e., filament, sheet, or disk) as constrained by laboratory measurements .

The complexity of Eq. () has historically led to roughly three different types of numerical solution methods: point-based methods, target-based spectral moment methods, and source-based spectral moment methods.

Point-based approaches such as the method approximate the number distribution function as a set of discrete points and attempt to solve Eq. () by interpolating the functional gain term. This approach is computationally intensive compared to other methods but is considered to be quite accurate and is often used to provide reference solutions for other methods e.g.,. The gain and loss terms are handled separately which leads to mass not being exactly conserved, although with enough points this lack of conservation can be mitigated somewhat.

Target-based spectral moment methods instead evolve bin moments (i.e., number and/or mass) and were originally developed by and and improved upon by and . In the moment-based approach, Eq. () is integrated for each k bin with the kernel xrdx from the left bin edge xk to the right bin edge xk+1. This transforms Eq. () into 2∂Mkr(z,t)∂t+∂Skr(z,t)dz=12∫m=xkxk+1mrdm∫x=0mC(m-x,x)n(m-x,z,t)n(x,z,t)dx-∫m=xkxk+1mrn(m,z,t)dm∫x=0∞C(m,x)n(x,z,t)dx+12∫m=xkxk+1mrdm∫x=0∞∫y=0∞B(x,y)P(m|x,y)n(x,z,t)n(y,z,t)dydx-∫m=xkxk+1mrn(m,z,t)dm∫y=0∞B(m,y)n(y,z,t)m+ydy∫x=0m+yxP(x|m,y)dx, where Mkr(z,t)≡∫m=xkxk+1mrn(m,z,t)dm is the rth mass moment of bin k and Skr(z,t)≡∫xkxk+1mrvt(m)n(m,z,t)dm is the bin's rth mass moment vertical flux. Often, Eq. () is used to generate ordinary differential equations for evolving the mass moment (r=1) and the number moment (r=0). While this integral transform increases the dimensionality by one, the advantage is that differential equations for evolving moments of the number distribution function can be represented, at least partially, in terms of the moments themselves. This can be seen schematically in Fig.  for collision-coalescence (panel a) and breakup (panel b). In Fig. a, each k bin of the target distribution is treated separately one-at-a-time. Each k bin integrates all i and j source bin regions that overlap with the k bin such that xk≤x+y≤xk+1. This can be seen in Fig. a by the red “Gain” region created by the x+y=xk and x+y=xk+1 diagonal lines. This collision-coalescence gain term for bin k can be numerically calculated by splitting the full integral region (red) into quadrilateral regions of all i and j bin interactions where the individual interacting source i and j bin amounts are shown as black hashes. The loss region for the k bin (blue) is simply a rectangular integration region created by the k (i) bin interaction with each j bin. The gain term is often very lengthy and suffers from the problem of closure, i.e. the subregions are expressed in terms of higher-order moments of the i and j bins. This has led to different closure techniques. For example, uses a nondimensional integral parameter to connect higher order moments to lower order moments whereby the error is small and the maximum possible error can be determined using Schwarz's inequality for a given bin resolution. Because of this error, mass conservation might not be completely satisfied, however the error is apparently minimal. These tricky integral regions are even more complicated when considering collisional breakup as shown in Fig. b where these quadrilateral gain and loss regions are represented as three dimensional volumes that are constrained and separated by the x+y=m and x=y+m planes. Other methods such as used by have simplified the lengthy gain and loss integral terms by using the so-called “fixed-pivot” method pioneered by and later by . In this approach, particles are redistributed by conserving mass and number through a linear interpolation between bin edges and the newly generate particles from the gain terms.

Source-based spectral moment methods are perhaps the most conceptually simple to understand and implement. In these approaches such as detailed in , each bin-pair interaction (i.e., each pair of i and j bin) and integral region are computed separately and then redistributed to the target distribution bins according to the gain integral regions for each target bin that map to the bin-pair interaction source region. These methods utilize Eulerian geometric mass grids as this ensures that the redistribution will always include a maximum of two bins: a k bin and k+1 bin. By considering each source bin-pair interaction, mass conservation for each bin-pair interaction can be ensured unlike the first two methods. Various source-based models treat the k and k+1 bin partitioning differently. uses an advection or “flux” scheme whereby the total mass of the i and j collection event are placed in the k bin and a certain fraction of this total mass is then transferred to the k+1 bin by calculating a fractional parameter that is analogous to a weighted Courant number. The method is even simpler in that it approximates the collection kernel as a constant such that each source bin-pair interaction provides a fully closed set of ordinary differential equations in terms of the mass and number moments of the interacting distribution bins. A time-dependent linear distribution is then fit between the k and k+1 bins in order to partition the total mass and number between each gain bin.

The source-based method of the Bin Integral Method with Gauss Quadrature (BIMGQ) model by provided a more rigorous and exact way to perform the k and k+1 bin partitioning compared to the linear flux method of and the improved version of as well as the linear discrete method of . found that the source bin-pair integral area regions can be calculated exactly through purely geometric considerations and a bilinear approximation of the collection kernel leads to theoretically polynomial terms for each source bin pair. In this way, the gain terms for the k or k+1 bin are expressed in terms of the full source region defined by the i and j bin interaction. Therefore, only the k or k+1 integration calculation is needed as the other bin's calculation can be determined by subtracting the other bin's contribution from the total rectangular source region. In order to simplify these integration calculations, used a Gaussian quadrature routine which, with enough nodes, would yield the exact polynomial values. further uses the two predicted moments to diagnose a subgrid linear distribution using the inversion formulas of . Thus, the model evolves both total k bin mass Mk and number Nk whereas and evolves the mass distribution function directly by transferring bin-pair interacting masses and then using this mass transfer to fit the new mass distribution function.

While classic Fortran-based collision-coalescence spectral bin solvers exist such as the TAU model and the scheme, there has recently been an adoption toward more accessible, modern languages. This transition toward modern accessibility is also reflected in 1D column models and coalescence-based microphysics solvers. For instance, Python-based packages such as LCM1D and PySDM use the Lagrangian super-droplet method to simulate coalescence. Similarly, Cloudy.jl from provides a Julia-based bulk microphysics scheme for simulating cloud microphysics and coalescence. Python-based packages such as the DustPy package from also simulate coagulation but for different disciplines such as astrophysics. For weather radar purposes, it is often not necessary to produce full 3D or 2D simulations when data is often displayed and analyzed in a time/height format such as for quasi-vertical profiles , range-defined quasi vertical profiles , columnar vertical profiles , Path-columnar vertical profiles . In these cases, it would be convenient for users to specify some initial distribution and then efficiently produce corresponding spectral distributions as well as microphysical and forward simulated radar variables representing mean vertical profiles. Thus, users could use such a simplified model to investigate plausible microphysical explanations for observed vertical radar profiles or for developing retrieval methods from such vertical profiles.

To this end, this paper details a new class-based Python package “BinMod1D” which provides an efficient and versatile spectral bin model along with useful plotting methods. The previous collision-coalescence and breakup methods discussed above often focus on a box model setup and only account for interactions within a single particle class. In contrast, BinMod1D can incorporate as many particle classes as desired as well as custom coalescence and breakup kernels and fragment distributions. Furthermore, the model can be run as a box model, a height-based steady-state model, or a full 1D time/height model. In particular, the steady-state versions of Eqs. () and () are functionally distinct from the box model and 1D column model in the sense that the tendency term is omitted. Therefore, the ability of BinMod1D to directly simulate steady-state vertical profiles drastically decreases the computational time compared to running the 1D column model to its steady state as done in several rainshaft studies e.g.,. The BinMod1D model is based upon the source-based approach of but with several computational optimizations. For example, the model heavily utilizes Numba which automatically parallelizes calculations for computationally intensive model runs. The model can also be run using the 2-moment (mass and number) subgrid linear distribution approach of or as a 1-moment (mass only) subgrid uniform distribution approach. The BinMod1D methodology is described in more detail in Sect.  along with the Python class architecture. Section  provides several box model, steady-state, and column model results derived from Jupyter notebook examples available in the BinMod1D v1.0.10 repository . Section  performs a computation speed and accuracy test using a realistic meteorological scenario for both two-moment and one-moment calculations. Finally, Sect.  discusses several potential uses of BinMod1D and likely future additions and improvements to the package.

Most of the results presented in this section are derived from Jupyter notebook examples: “BM1D_analytical_examples.ipynb”, “BM1D_steady_state_examples.ipynb”, and “BM1D_full_1D_examples.ipynb”, which are provided in the BinMod1D repository . Users can execute these examples locally or within a pre-configured Binder environment which can be accessed through the project's documentation interface in .

Timing tests are performed on a workstation laptop equipped with a 12th generation Intel(R) Core i7-12850HX processor (2.1 GHz and 16 cores) and 64 GB of RAM running in a standard Python environment. These specifications represent standard high-end consumer hardware, thereby demonstrating that the model can be effectively used without the need for specialized high-performance computing (HPC) resources.

These timing and convergence tests use the box model capability of BinMod1D to simulate a transition from continental cloud droplets to moderate rain with Z≈40 dBZ. This is similar to simulations performed by and . However, unlike those simulations, the BinMod1D simulations include both collision-coalescence and collisional breakup. Here, the precipitation type is considered to be rain with one category and the fall speed parameterization is used rather than the default power-law parameterization. The Long kernel (kernel='Long') is used where the collision efficiency

The kernel='Long' option actually modifies the collision kernel K(x,y) rather than Ecol to keep Ecol as a constant parameter. In these cases, Ecol can be set to unity.

BinMod1D is a simple, economic spectral bin microphysics model that should be accessible for those familiar with the Python programming language. Other similar Python packages like LCM1D and PySDM use Lagrangian depictions of the PSD evolution rather than the typical Eulerian description given by traditional spectral bin microphysics schemes. In this way, BinMod1D represents a much more traditional approach that is more similar to the widely-used Fast Spectral Bin Microphysics scheme FSBM; which is available in the Fortran-based Weather Research and Forecasting (WRF) model. BinMod1D provides an expedient way to not only generate box model or 1D column model PSD evolution due to coalescence and/or breakup, but also to easily incorporate various habit parameters, numbers of distributions, and interaction kernels. Additionally, BinMod1D, unlike other similar Python packages, allows for direct, explicit simulation of steady-state vertical radar profiles. This article provides only a small set of examples for using the model. However, due to its flexibility, BinMod1D can be customized for a variety of different purposes. For example, users could use BinMod1D to perform in-situ-type bin model simulations similar to where actual in-situ-derived particle size distributions are used to initialize the bin model and investigate the theoretical effects of aggregation (and/or breakup) compared to measured distributions taken from such Lagrangian aircraft spirals. Similarly, users could also develop theoretical particle size distributions valid for laboratory studies such as done in .

Figure  presents another possible application: retrieval methodology validation. In this example, BinMod1D is run in the steady-state mode to produce synthetic snow size distributions in height. Here, sbin=6 with 180 bins in order to produce a high resolution set of PSDs. This simulation uses an initial exponential distribution with parameters Nt,0=50 L⁻¹ and Dm,0=1.25 mm which produces an initial reflectivity of about 20 dBZ. Es=0.35 is used to simulate the explicit steady-state PSD evolution in height due only to aggregation. This increases the reflectivity to approximately 30 dBZ at the surface. The Z/Kdp (B20) and the Z/ZDR/KDP (RZ19) power-law retrievals are then used to retrieve Nt, Dm, IWC, and R solely from the BinMod1D polarimetric radar height signatures see Table 1 fromfor these retrieval equations. In this way, these comparisons shows the limitations of these retrieval methods due to using a fixed distribution shape. The BinMod1D simulation and retrievals assumes the same snow parameters as used in to derive their retrievals where the snow aspect ratios are 0.65, σ=0°, the density-size power-law relation is ρ(D)=0.178D-1, and the fall speed-size relation is vt(D)=0.81D0.15. Both sets of retrievals assume μ=0 for all heights. Initially, the retrieved size distribution parameters (Fig. a, b) are within 5 L⁻¹ and 0.05 mm of the initial BinMod1D exponential distribution Nt and Dm, respectively. The initial differences between the BinMod1D microphysical variables are due to the capping of the density at 0.92 g cm⁻³ at small sizes which the analytical retrieval formulas do not consider (for the retrievals to be simple power-law formulas, ρ(D→0)=∞). Both B20 and RZ19 demonstrate the same general behavior of Nt(z) and Dm(z) from the explicit BinMod1D steady-state simulation. However, both retrievals underestimate the increase in Dm due to aggregation toward ground by nearly 1.0 mm. Figure  panels (i) through (k) shows that this is due to the retrievals tendency to approximate the quasi-exponential behavior in the millimeter size range but underestimate the size distribution of large aggregates present in the centimeter range. These sparse but large aggregates that generate in the BinMod1D simulation overall heavily weights the Dm calculation. Both B20 and RZ19 overestimate the snow rates particularly at the surface where both retrievals overestimate the size distribution in the mid-millimeter range (Fig. k). Figure c show that B20 retrieves a near constant IWC profile whereas RZ19 retrieves an IWC profiles that increases toward ground unlike the slightly decreasing IWC profile from BinMod1D. Overall, while the retrievals exhibit the correct overall behavior, these retrievals are still fundamentally limited by their inability to account for a changing distribution shape due to aggregation. This is one example where the reference solution provided by BinMod1D could be used to improve the retrieval equations of B20, RZ19, and others. Similarly, BinMod1D reference solutions can also be used to train retrieval-based artificial intelligence models.

There are a number of planned future improvements to the BinMod1D package including but not limited to

T-matrix radar forward simulation calculations.