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Hail trajectory modeling is a popular tool to explore how environment and storm characteristics allow or prohibit large hail growth. However, trajectory models are complex and computationally expensive: storm dynamics relevant to hail growth are inextricably linked such that ``cause’' and ``correlation” are difficult to distinguish. Therefore, we develop a novel hail trajectory model that can be used to untangle hail growth processes. Toward this end, we explore the vertical structure of vertical velocity and liquid water content in updrafts and define analytic functions that approximate the thermodynamic prediction of these quantities. These analytic profiles are used, along with a temporal updraft-pass parameterization to define a 2D updraft (defined in height and time) in which hailstones can grow. Hail growth in this 2D updraft is fully defined by a set of 16 scalar parameters that act as turnable knobs to produce unique hail trajectories. This article is Part I of a series using this modeling framework to explore the nature of hail growth. Here, we define the model and test its ability to produce realistic hail trajectories and hail sizes through a Monte Carlo simulation with physical couplings maintained. The size distribution from 1 billion simulated trajectories is exponential and has maximum hail size of 25.7 cm. Stochasticity in the model’s representation of hail fall speed and cross-sectional area is explored and produces some variability in the resulting hailstone sizes. The model produced and evaluated here will be used in further studies to identify how environment, updraft, and hail embryo characteristics individually impact hail growth. 

This is Part II of a multi-part series exploring the fundamental nature of hail growth through a toy model developed in Part I. The toy model uniquely parameterizes all hail growth processes by single-valued parameters with great computational efficiency. The parameters are uncoupled so that environment, storm, and hail embryo characteristics, and their impact on hail growth, can be studied independently---a task 3D trajectory models cannot perform because of their highly coupled nature. Three Monte Carlo simulations were run to compare hail growth from small and large hail embryos, and coupled and uncoupled model parameters. Hail with maximum dimension ($$D$$) $$\le25.71$$ cm grew in the physically coupled small-embryo simulation, $$D\le33.59$$ cm hail grew in the physically coupled large-embryo simulation, and $$D\le44.97$$ cm hail grew in the uncoupled large-embryo simulation. The largest hailstones from the three Monte Carlo simulations took similar trajectories, accumulating a large proportion of their mass both while suspended and during their fall. Analysis of model parameters corroborate current hail growth theory, indicating three necessary ingredients for large hail: (1) a favorable embryo size and location, (2) a long residence time in a water-rich updraft, and (3) a balance between updraft vertical velocity and hailstone fall speed. The sensitivity of hail size to these parameters is analyzed: a hailstone's potential size is limited by its updraft-pass duration and the available amount of supercooled liquid water, but hail size is most sensitive to the balance between its fall speed and its encountered updraft vertical velocity.