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Figure 1. Thunderstorm systems from different locations and seasons, from left to right: tropical thunderstorms in Florida and New Mexico, a winter thunderstorm in Japan, and an atmospheric river in California (first row). Typical thunderstorm types, including single-cell, multi-cell, squall line, and supercell, are shown in the second row. Each image represents a unique thunderstorm system observed in meteorological studies.
\Description

Illustrations of eight thunderstorm systems: tropical thunderstorms, winter thunderstorm, atmospheric river, single-cell, multi-cell, squall line, and supercell. Each image represents a unique thunderstorm system observed in meteorological studies.

Thunderscapes: Simulating the Dynamics of Mesoscale Convective System

TIANCHEN HAO
Abstract.

A Mesoscale Convective System (MCS) is a collection of thunderstorms that function as a system, representing a widely discussed phenomenon in both the natural sciences and visual effects industries, and embodying the untamed forces of nature.In this paper, we present the first efficient, physically based mesoscale thunderstorms simulation model that integrates Grabowski-style cloud microphysics with hydrometeor electrification processes. Our model simulates thunderclouds development and lightning flashes within a unified meteorological framework, providing a realistic and efficient approach for graphical applications. By incorporating key physical principles, it effectively links cloud formation, electrification, and lightning generation. The simulation also encompasses various thunderstorm types and their corresponding lightning activities. For more details, see the dynamic video: Thunderscapes Video.

Thunderstorm Modeling and Simulation,Fluid Simulation, Cloud Simulation,Lightning Simulation,Weather Simulation,Atmospheric Microphysics,Atmospheric Electrification
ccs: Computing methodologies Physical simulation

1. INTRODUCTION

Thunderstorms represent the wild power of nature and are a common atmospheric element in visual effects (VFX) industry. Notable works, such as Horizon Forbidden West: Burning Shores and Ghost of Tsushima(see Figure 2), demonstrate the importance of realistic atmospheric effects. Modern applications demand tools for efficient and realistic simulation of thunderstorms. However, current general purpose VFX software, such as Houdini and Maya, lacks standardized toolkits specifically designed for creating atmospheric phenomena like thunderstorms.

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(a) Thunderstorm from *Ghost of Tsushima*. ©2020 Sony Interactive Entertainment LLC.
\Description

A scenic image from the video game Ghost of Tsushima showing dramatic landscapes and warrior scenes.

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(b) Thunderstorm from *Horizon Forbidden West: Burning Shores*. ©2023 Sony Interactive Entertainment Europe.
\Description

A scene from the video game Horizon Forbidden West: Burning Shores, showcasing futuristic landscapes and action.

Figure 2. Comparison of atmospheric visual styles between *Ghost of Tsushima* and *Horizon Forbidden West: Burning Shores*.

Recent advances in computer graphics research have increasingly focused on cloud dynamics, incorporating atmospheric microphysical processes (Hädrich et al., 2020; Herrera et al., 2021; Amador Herrera et al., 2024). However, the critical role of atmospheric electrification in cloud development remains underexplored. This study aims to address this gap by integrating the electrification process, thus enhancing the consistency and realism of phenomena observed during thunderstorm events.

This paper presents an efficient, physically based method for simulating thunderstorm development. By coupling cloud microphysics with atmospheric hydrometeor electrification, our approach captures essential processes that contribute to the consistent formation of mesoscale thunderstorm phenomena.

The key contributions include:

  1. (1)

    We present a comprehensive framework for computing thunderstorm microphysics in the atmosphere, incorporating the transport of cloud hydrometeors and their electrification processes.

  2. (2)

    We introduce a lightweight parameterization that enables users to efficiently simulate various types of thunderstorms, along with their corresponding lightning effects.

  3. (3)

    We validate our simulation using different meteorological datasets to ensure the generation of physically credible and visually accurate atmospheric effects.

2. RELATED WORK

2.1. Simulating Thunderstorms in Computer Graphics

The simulation of atmospheric phenomena such as thunderstorms has been extensively explored through a variety of computational methods. Webanck et al. (Webanck et al., 2018) proposed a procedural approach for generating cloudscapes, while Miyazaki et al. (Miyazaki et al., 2002) simulated cumulus clouds by coupling computational fluid dynamics (CFD) with fundamental water transport equations. Ferreira et al. (Ferreira Barbosa et al., 2015) and Zhang et al. (Zhang et al., 2020) utilized position-based fluids (PBF) for adaptive cloud simulations. Smoothed Particle Hydrodynamics (SPH) techniques, as demonstrated by Goswami and Neyret (Goswami and Neyret, 2017), focus on real-time simulations of convective clouds. Additionally, Vimont et al. (Vimont et al., 2020) proposed a hybrid, 2D-layered atmospheric model to simulate mesoscale skyscapes.

Some studies have specifically focused on the simulation of volcanic cloud dynamics. Lastic et al. (Lastic et al., 2022) employed Lagrangian dynamics to simulate volcanic plumes and pyroclastic flows, while Pretorius et al. (Pretorius et al., 2024) integrated volcanic eruptions with atmospheric simulations to produce coherent skyscapes.

In the domain of lightning simulation, Reed and Wyvill (Reed and Wyvill, 1994), Kim and Lin (Kim and Lin, 2007), and Yun et al. (Yun et al., 2017) developed methods for the efficient development of lightning branches, contributing to a more realistic representation of the method of electrical discharges.

Recently, more sophisticated microphysical schemes from atmospheric science have been incorporated into computer graphics research. Garcia-Dorado et al. (Garcia-Dorado et al., 2017) and Hädrich et al. (Hädrich et al., 2020) adopted the classic Kessler warm cloud microphysics scheme (Kessler, 1969) in cloud simulations. Herrera et al. (Herrera et al., 2021) extended cloud simulations to include multiphase cloud dynamics. Amador Herrera et al. (Amador Herrera et al., 2024) developed a framework to simulate hurricane and tornado dynamics.

2.2. Thunderstorm models in atmospheric sciences

Thunderstorm microphysics and electrification have been widely discussed topics in the field of atmospheric science.

Kessler (Kessler, 1969) proposed a fundamental framework for the distribution and continuity of water substance in atmospheric circulations, which remains influential in the parameterization of the microphysics of warm clouds.

One notable development in cloud microphysics is the work by Grabowski (Grabowski, 1998), who introduced an extended warm cloud microphysics scheme for large-scale tropical circulations. His method divides the parameterization of warm and cold clouds using a temperature interpolation scheme, a significant inspiration for our approach.

In terms of thunderstorm electrification, Solomon et al. (Solomon et al., 2005) introduced a 1.5-dimensional explicit microphysics thunderstorm model that incorporates a lightning parameterization, addressing key aspects of thunderstorm electrification. Furthermore, Mansell et al. (Mansell et al., 2002) simulated three-dimensional branched lightning in a numerical thunderstorm model, providing insights on the complex activities of lightning formation. Barthe and Pinty (Barthe and Pinty, 2007) further advanced the field by simulating a supercell storm using a three-dimensional mesoscale model with an explicit lightning flash scheme, capturing the lightning activities specific to supercell thunderstorms.Mansell et al. (Mansell et al., 2010) also examined the electrification of small thunderstorms using a two-moment bulk microphysics scheme, extending the understanding of lightning activity in small multicell thunderstorms.

Our work focuses on the efficient simulation of common thunderstorm types, based on the standard categories in Mesoscale Convective Systems (MCS). According to the National Severe Storms Laboratory (NSSL)111https://www.nssl.noaa.gov/education/svrwx101/thunderstorms/types/, these types include single cell, multi-cell, squall line, and supercell thunderstorms. Additionally, our simulation includes common phenomena generated by thundercloud electrification, such as cloud-to-ground (CG) and intra-cloud (IC) lightning222https://www.nssl.noaa.gov/education/svrwx101/lightning/types/.

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Figure 3. Schematic illustration of our thunderstorm microphysics scheme for hydrometeor phase transitions and the electrification process. The details will be further discussed in Chapter 4. (I) Blue frame: the transport of water substances during thundercloud development; (II)Orange frame: the accumulation of charge density due to the collision-coalescence of hydrometeors; (III) Red frame: the static charge, once it exceeds a given electric field threshold, triggers the lightning discharge process; (IV) Blue arrow: the neutralization of hydrometeors following the lightning flash.
\Description

A diagram illustrating the stages of hydrometeor phase transitions and electrification during thunderstorm development.

3. OVERVIEW

The primary motivation of our approach is to create consistent atmospheric phenomena during thunderstorms, particularly focusing on thundercloud development and dissipation, along with lightning flashes resulting from the thundercloud electrification processes,as illustrated in Figure 3. By simulating both the microphysics of thunderstorms and their electrodynamic properties, we aim to achieve realistic, efficient weather simulations suitable for graphical applications.

Our model integrates key physical principles to simulate thunderstorm behavior at a mesoscale level. The thunderstorm microphysics component models atmospheric dynamics by coupling a Grabowski-style extended warm cloud microphysics scheme with hydrometeor electrification processing. This approach ensures that the microphysical processes driving cloud formation and growth are consistently linked to the electrification necessary for lightning generation.

The key atmospheric quantities driving our model include vapor, cloud water, ice, precipitated rain , precipitated snow , static charge (electric charge distribution within the thundercloud, crucial for lightning formation), and lightning (the flash itself, including both its onset and development).

The validation of our model involves presenting spatially simulated thundercloud structures alongside meteorological characteristics and comparing the temporally simulated results with real-world weather data. This includes analyzing cloud fraction profiles to assess cloud formation and structure, tracking the evolution of cloud coverage against real-time data from national weather services333https://www.visualcrossing.com/weather/weather-data-services/, and evaluating the temporal variation in lightning flash rates.

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Figure 4. Illustration of our thunderstorm microphysics scheme, which integrates the Grabowski-style extended warm cloud microphysics with the hydrometeor electrification process.
\Description

A schematic representation of the thunderstorm microphysics scheme, coupling the extended warm cloud microphysics with the electrification process of hydrometeors.

4. METHODOLOGY

Our microphysics model, shown in Figure 4, illustrates the interactions among hydrometeor phase transitions and the electrification mechanisms that lead to lightning. Key processes include the condensation of water vapor into droplets and ice crystals, which subsequently form precipitation through autoconversion and accretion. Evaporation recycles hydrometeors back into the vapor phase, while collisions and coalescence facilitate charge separation. Lightning occurs when the electric field strength exceeds a critical threshold, resulting in the discharge and redistribution of accumulated charge. This feedback loop provides a conceptual framework for understanding thunderstorm development within a mesoscale convective system.

4.1. Cloud Microphysics

The fundamental warm-cloud microphysics equations describe the evolution of potential temperature, water vapor, cloud condensate, and precipitation:

(1) DθDt=LvθcpT(CONDDIFF)\displaystyle\frac{D\theta}{Dt}=\frac{L_{v}\theta}{c_{p}T}(COND-DIFF)
(2) DqvDt=COND+DIFF\displaystyle\frac{Dq_{v}}{Dt}=-COND+DIFF
(3) DqcDt=CONDAUTCACCR\displaystyle\frac{Dq_{c}}{Dt}=COND-AUTC-ACCR
(4) DqpDt=AUTC+ACCRDIFF\displaystyle\frac{Dq_{p}}{Dt}=AUTC+ACCR-DIFF

Here, θ\theta represents potential temperature, LvL_{v} is the latent heat of condensation or evaporation, and cpc_{p} is the specific heat capacity at constant pressure. The variables qvq_{v}, qcq_{c}, and qpq_{p} denote the mixing ratios of water vapor, cloud condensate, and precipitation, respectively. The terms CONDCOND, DIFFDIFF, AUTCAUTC, and ACCRACCR represent the condensation rate, diffusional growth rate of precipitation, autoconversion rate, and accretion rate, respectively.

We adopt the equilibrium approach proposed by Grabowski, incorporating a temperature-dependent factor α\alpha to differentiate warm and cold cloud microphysics. Warm clouds dominate above 0C0^{\circ}\mathrm{C} and cold clouds below 20C-20^{\circ}\mathrm{C}, with a linear interpolation for intermediate temperatures:

(5) qvs=αqvw+(1α)qvi\displaystyle q_{vs}=\alpha q_{vw}+(1-\alpha)q_{vi}
(6) qw=αqc,qi=(1α)qc\displaystyle q_{w}=\alpha q_{c},\,q_{i}=(1-\alpha)q_{c}
(7) qr=αqp,qs=(1α)qp\displaystyle q_{r}=\alpha q_{p},\,q_{s}=(1-\alpha)q_{p}

Here, qvsq_{vs} represents the total vapor saturation, combining vapor saturation over water (qvwq_{vw}) and ice (qviq_{vi}); qwq_{w} and qiq_{i} denote cloud water and ice, expressed as fractions of the cloud-ice mixture (qcq_{c}); qrq_{r} and qsq_{s} represent rain and snow, derived as fractions of the rain-snow mixture (qpq_{p}).

The transport equations for rain and snow processes include separate contributions from diffusional growth (DIFFDIFF), autoconversion (AUTCAUTC), and accretion (ACCRACCR) rates:

(8) DIFF=DIFFr+DIFFs\displaystyle DIFF=DIFF_{r}+DIFF_{s}
(9) AUTC=AUTCr+AUTCs\displaystyle AUTC=AUTC_{r}+AUTC_{s}
(10) ACCR=ACCRr+ACCRs\displaystyle ACCR=ACCR_{r}+ACCR_{s}

The terminal velocity vpv_{p} of precipitation is modeled as a weighted combination of rain (vrv_{r}) and snow (vsv_{s}) velocities, with typical values vr=10m/sv_{r}=-10\,\mathrm{m/s} for rain and vs=2m/sv_{s}=-2\,\mathrm{m/s}:

(11) vp=αvr+(1α)vs\displaystyle v_{p}=\alpha v_{r}+(1-\alpha)v_{s}

The saturation vapor mixing ratio qvsq_{vs} combines contributions from vapor saturation over water and ice, weighted by α\alpha(Yau and Rogers, 1996):

(12) qvs=380.16p[αexp(17.67TT+243.50)+(1α)exp(24.46TT+272.62)]q_{vs}=\frac{380.16}{p}\left[\alpha\exp\left(\frac{17.67\cdot T}{T+243.50}\right)+(1-\alpha)\exp\left(\frac{24.46\cdot T}{T+272.62}\right)\right]

Here, pp represents the pressure, and TT denotes the temperature. The saturation vapor is modeled as an exponential distribution for both liquid water and ice.

The diffusional growth rate DIFFDIFF accounts for condensation and evaporation rates, integrating water and ice contributions (Dudhia, 1989),where βr,βs\beta_{r},\beta_{s} are phase-specific coefficients:

(13) DIFF=βr(αqvsqv)++βs((1α)qvsqv)+DIFF=\beta_{r}\cdot\left(\alpha\cdot q_{vs}-q_{v}\right)^{+}+\beta_{s}\cdot\left((1-\alpha)\cdot q_{vs}-q_{v}\right)^{+}

The autoconversion rate AUTCAUTC describes cloud condensate aggregation into precipitation (Kessler, 1995; Lin et al., 1983):

(14) AUTC=βr(αqc103)++βse0.025T((1α)qc103)+AUTC=\beta_{r}\cdot(\alpha\cdot q_{c}-10^{-3})^{+}+\beta_{s}\cdot e^{0.025T}\cdot((1-\alpha)\cdot q_{c}-10^{-3})^{+}

The accretion rate ACCRACCR models the collection of cloud condensate by precipitation particles (Morrison et al., 2015):

(15) ACCR=qcqr(βrα2+βs(1α)2)ACCR=q_{c}\cdot q_{r}\cdot\left(\beta_{r}\cdot\alpha^{2}+\beta_{s}\cdot(1-\alpha)^{2}\right)

4.2. Electrification

Based on the Reynolds-Brook theory of thunderstorm electrification (Latham and Miller, 1965), the fair-weather electric field (EE) induces opposite charges on precipitation particles. Collisions between particles, influenced by their velocities and directions, result in partial charge neutralization and a residual net charge. This theory underpins the following mathematical model for electrification, with charge density (ρ\rho) defined as:

(16) ρ=kα(1α)(qc2vcqp2vp)e10qv\rho=k\alpha(1-\alpha)\left(q_{c}^{2}v_{c}-q_{p}^{2}v_{p}\right)e^{-10q_{v}}

where kk is a user-defined constant, vcv_{c} and vpv_{p} are their respective velocities of cloud-ice mixture and rain-snow mixture.

The threshold electric field (EtrE_{tr}) required for lightning initiation depends on the altitude (ρA(h)\rho_{A}(h)) as defined by (Marshall et al., 1995):

(17) Etr=±167ρA(h)whereρA(h)=1.208exp(h8.4)E_{tr}=\pm 167\,\rho_{A}(h)\quad\text{where}\quad\rho_{A}(h)=1.208\exp\left(\frac{-h}{8.4}\right)

Charge neutralization processes, occurring after a lightning discharge as described by (Barthe and Pinty, 2007), are governed by the following relationship:

(18) δρ={±(|ρ|ρexcess),if |ρ|>ρexcess0,if |ρ|ρexcess\delta\rho=\begin{cases}\pm\left(|\rho|-\rho_{\text{excess}}\right),&\text{if }|\rho|>\rho_{\text{excess}}\\ 0,&\text{if }|\rho|\leq\rho_{\text{excess}}\end{cases}

Here, δρ\delta\rho represents the net charge change, and ρexcess\rho_{\text{excess}} denotes the threshold for excess charge density. The total neutralized charge density, accounting for the collective contributions of lightning growth points, is expressed as:

(19) δρneut=1N|(2α1)qcqp|nδρ\delta\rho_{\text{neut}}=\frac{1}{N}\left|(2\alpha-1)q_{c}q_{p}\right|\sum_{n}\delta\rho

Finally, a suppression factor (SS) is introduced to modulate lightning activity frequency as thunderstorms dissipate. This factor ensures consistent lightning activity during the lifecycle of thunderstorms:

(20) S=11+k|δρneut|S=\frac{1}{1+k\cdot|\delta\rho_{\text{neut}}|}

4.3. Atmospheric Background

Our atmospheric background is based on the theory proposed by Hädrich et al. (Hädrich et al., 2020). Specifically, we assume that the atmosphere is initially electroneutral, with the charge density, denoted as ρ\rho, being zero.

The isentropic exponent γth\gamma_{\text{th}} for the air-water mixture (Anderson, 1990) is calculated as a weighted average of the vapor-specific exponent γvapor\gamma_{\text{vapor}} and the air-specific exponent γair\gamma_{\text{air}}, as shown below:

(21) γth=Yvaporγvapor+(1Yvapor)γair\gamma_{\text{th}}=Y_{\text{vapor}}\gamma_{\text{vapor}}+\left(1-Y_{\text{vapor}}\right)\gamma_{\text{air}}

where YvaporY_{\text{vapor}} represents the mass fraction of water vapor in the air. The values of γair\gamma_{\text{air}} and γvapor\gamma_{\text{vapor}} are taken as 1.4 and 1.33, respectively, based on standard thermodynamic properties of dry air and water vapor.

The atmospheric temperature profile (Atmosphere, 1975), T(h)T(h), is modeled by a piecewise function that accounts for the lapse rate, including the effect of the inversion layer at a height h1h_{1}. The temperature at a given altitude hh is expressed as:

(22) T(h)={T0+Γ0h,0hh1T0+Γ0h1+Γ1(hh1),h1hT(h)=\begin{cases}T_{0}+\Gamma_{0}h,&0\leq h\leq h_{1}\\ T_{0}+\Gamma_{0}h_{1}+\Gamma_{1}(h-h_{1}),&h_{1}\leq h\end{cases}

where T0T_{0} is the base temperature at sea level, Γ0\Gamma_{0} and Γ1\Gamma_{1} represent the lapse rates in the lower and upper layers.

The atmospheric pressure profile, p(h)p(h), is derived from the hydrostatic equation (Houze Jr, 2014), considering the effect of gravity and the ideal gas law. It is given by:

(23) p(h)=p0(1ΓhT0)gRT0p(h)=p_{0}\left(1-\frac{\Gamma h}{T_{0}}\right)^{\frac{g}{RT_{0}}}

where p0p_{0} is the pressure at sea level, gg is the acceleration due to gravity, and RR is the specific gas constant.

To model the thermodynamic properties of humid air, we calculate the average molar mass of the air-water mixture M𝑡ℎ\mathit{M}_{\mathit{th}}. This is given by:

(24) M𝑡ℎ=XvaporM𝑤𝑎𝑡𝑒𝑟+(1Xvapor)M𝑎𝑖𝑟\mathit{M}_{\mathit{th}}=X_{\text{vapor}}\mathit{M}_{\mathit{water}}+\left(1-X_{\text{vapor}}\right)\mathit{M}_{\mathit{air}}

where XvaporX_{\text{vapor}} is the mole fraction of water vapor, and M𝑤𝑎𝑡𝑒𝑟\mathit{M}_{\mathit{water}} and M𝑎𝑖𝑟\mathit{M}_{\mathit{air}} are the molar masses of water (18.02 g/mol) and dry air (28.96 g/mol), respectively.

The mass fraction of water vapor in the humid air, YvaporY_{\text{vapor}}, is related to the mole fraction XvaporX_{\text{vapor}} by:

(25) Yvapor=XvaporM𝑤𝑎𝑡𝑒𝑟M𝑡ℎY_{\text{vapor}}=X_{\text{vapor}}\frac{\mathit{M}_{\mathit{water}}}{\mathit{M}_{\mathit{th}}}

The temperature of the air in the atmosphere can also be related to pressure changes through the isentropic relation, which governs the temperature Tth(h)T_{\mathrm{th}}(h) at height hh in terms of the pressure profile. This is given by:

(26) Tth(h)=T0(p(h)p0)γth1γthT_{\mathrm{th}}(h)=T_{0}\left(\frac{p(h)}{p_{0}}\right)^{\frac{\gamma_{\mathrm{th}}-1}{\gamma_{\mathrm{th}}}}

Finally, the buoyancy force, which drives the upward movement of thundercloud, is calculated based on Archimedes’ principle and Newton’s second law. The buoyancy force B(h)B(h) at height hh is given by:

(27) B(h)=g(M𝑎𝑖𝑟M𝑡ℎT𝑡ℎ(h)T𝑎𝑖𝑟(h)1)B(h)=g\left(\frac{\mathit{M}_{\mathit{air}}}{\mathit{M}_{\mathit{th}}}\frac{T_{\mathit{th}}(h)}{T_{\mathit{air}}(h)}-1\right)

4.4. Electrodynamics

The electrodynamic behavior of the atmosphere is governed by Maxwell’s equations(Ma et al., 1998), which define the relationships between the electric field 𝐄\mathbf{E}, charge density ρ\rho, and electric potential VV. Gauss’s law describes how the divergence of 𝐄\mathbf{E} is proportional to ρ\rho:

(28) 𝐄=ρϵ0\nabla\cdot\mathbf{E}=\frac{\rho}{\epsilon_{0}}

where ϵ0\epsilon_{0} is the permittivity of free space. The electric field is related to the potential VV through:

(29) 𝐄=V\mathbf{E}=-\nabla V

indicating that 𝐄\mathbf{E} points in the direction of decreasing potential. Together, these equations describe the electrostatic field and potential in regions with charge density.

To analyze the electric potential in the atmosphere, we solve the Poisson equation, a fundamental equation derived from Gauss’s law in electrostatics. This equation describes the simplified spatial variation of the potential VV due to charge density ρ\rho(Kim and Lin, 2007):

(30) 2V=4πρ\nabla^{2}V=-4\pi\rho

For simulating lightning discharges, the Dielectric Breakdown Model (DBM) is employed(Niemeyer et al., 1984). The probability of a discharge at a specific point ii depends on the electric potential ViV_{i}, expressed as:

(31) pi=(Vi)ηj=1n(Vj)ηp_{i}=\frac{\left(V_{i}\right)^{\eta}}{\sum_{j=1}^{n}\left(V_{j}\right)^{\eta}}

Here, η\eta is a parameter controlling the spatial concentration of the discharge(Kim and Lin, 2007). Larger values of η\eta bias the discharge toward regions of higher potential, while smaller values allow for more diffusive propagation.

4.5. Fluid Dynamics

The motion of atmospheric fluids is described by the Navier-Stokes equations, which represent the conservation of momentum in a fluid medium (Wendt, 2008). These equations capture the effects of inertial forces, pressure gradients, viscosity, and external forces:

(32) (ρ𝐮)t+(ρ𝐮𝐮)=p+ρ(μ2𝐮+𝐛+𝐟),𝐮=0\frac{\partial(\rho\mathbf{u})}{\partial t}+\nabla\cdot(\rho\mathbf{u}\mathbf{u})=-\nabla p+\rho\left(\mu\nabla^{2}\mathbf{u}+\mathbf{b}+\mathbf{f}\right),\quad\nabla\cdot\mathbf{u}=0

Here, ρ\rho is the fluid density, 𝐮\mathbf{u} the velocity field, pp the pressure, μ\mu the dynamic viscosity, 𝐛\mathbf{b} the body force (e.g., gravity), and 𝐟\mathbf{f} external forces. The incompressibility condition 𝐮=0\nabla\cdot\mathbf{u}=0 ensures mass conservation for incompressible flows.

5. ALGORITHMICS

Algorithm 1 Thunderstorm Development Algorithm.
1:Input: Current MCS state (θ,p,ρ,u,qv,qc,qp)(\theta,p,\rho,u,q_{v},q_{c},q_{p}).
2:Output: Updated MCS state.
3:procedure 
4:     θ,p,ρ\theta,p,\rho\leftarrow Update atmospheric background conditions Eqs.(22–23)
5:     uu\leftarrow Advect and diffuse velocity field
6:     bb\leftarrow Compute thermal buoyancy Eq.(27)
7:     uu+b+fw+fvu\leftarrow u+b+f_{w}+f_{v} \triangleright Apply buoyancy, wind, vorticity confinement
8:     qv,qc,qpq_{v},q_{c},q_{p}\leftarrow Advect hydrometeor quantities
9:     uu\leftarrow Pressure projection
10:     qv,qc,qp,θ,ρq_{v},q_{c},q_{p},\theta,\rho\leftarrow Thundercloud microphysics update Eqs.(1–15)
11:     if Sρ>EtrS\cdot\rho>E_{tr} then \triangleright Electrification process Eq.(16–18)
12:         V,ρneutV,\rho_{\text{neut}}\leftarrow lightning discharge Eqs.(30–31),(19)
13:         SρS\cdot\rho\leftarrow Apply lightning neutralization Eq.(20)
14:     end if
15:end procedure

The theoretical framework discussed in the previous section is translated into a numerical procedure, as outlined in Algorithm 1. Figure 3 visualizes the key interrelationships governing thunderstorm development. We first present our methodology for simulating the life cycle of a Mesoscale Convective System (MCS), followed by an explanation of the dynamics of lightning channel growth triggered when the electric field exceeds a defined threshold.

5.1. Mesoscale Convective System Cycle

The input fields for the MCS system, depicted in Figure 5, serve as the foundation for initializing state variables, including potential temperature (θ\theta), pressure (pp), charge density (ρ\rho), velocity field (uu), and hydrometeor quantities (qvq_{v}, qcq_{c}, qpq_{p}).The solver updates these variables iteratively, producing an output that represents the evolved state of the MCS, incorporating changes driven by thundercloud microphysics and the electrification processes.

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(a)
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(b)
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(c)
Figure 5. Different types of sources used as input for our MCS solver,temperature and vapor field are sampled from perlin noise patterns. (a) Height field derived from satellite data, (b) ground temperature distribution, (c) vapor field distribution.

The procedure begins by updating the atmospheric background conditions,through governing equations (Eqs. 22–23).

The velocity field is then advected and diffused.Thermal buoyancy is computed (Eq. 27) and integrated into the velocity field, alongside contributions from wind and vorticity confinement forces.

Hydrometeor quantities (qvq_{v}, qcq_{c}, qpq_{p}) are transported according to the evolving velocity field through advection. Hydrometeor quantities (qvq_{v}, qcq_{c}, qpq_{p}) are transported through advection based on the evolving velocity field. For efficient pressure projection, we employ the compact Poisson filter scheme (Rabbani et al., 2022), a GPU-friendly method for solving large-scale sparse linear systems. Cloud microphysics are subsequently updated, incorporating processes such as condensation, evaporation, and precipitation, governed by parameterized equations (Eqs. 1–15).

Electrification is initiated when the charge density, modulated by the suppress factor (SρS\cdot\rho), exceeds the electric field threshold (EtrE_{tr}, Eqs. 16–18). This triggers lightning discharges, which are detailed in the next subsection. Following lightning activity, the neutralized charge density (ρneut\rho_{\text{neut}}, Eqs. 30–31, 19) is updated, redistributing charge density (SρS\cdot\rho, Eq. 20). The suppress factor is recalibrated to reflect the diminished likelihood of subsequent lightning events as the system dissipates.

5.2. Lightning Channel Growth

The growth of the lightning channel begins at a trigger point, randomly selected from locations where the electric field exceeds a predefined threshold. Inspired by the DBM model proposed by (Kim and Lin, 2007), the lightning channel stops growing once it connects to a positively charged object. This approach not only enhances flexibility in controlling the final structure of the lightning channel but also improves computational efficiency. The positive charge is distributed along the top of the height field.

The solution is computed on a volumetric grid, and the results are subsequently converted from grid points to geometric points. To create a more natural branching structure, a random dithering technique is applied to these points. The points are then connected to form the final geometry of the lightning branches.

6. VISUALIZATION

Our tool is implemented as a HDA(Houdini Digital Asset) using Houdini 20.0.625’s microsolver framework with OpenCL for GPU acceleration. The hardware setup includes NVIDIA® GeForce® RTX A6000 GPU,13th Gen Intel® Core™ i9-13900 processor, and 128GB of RAM.

The simulation results demonstrate various thunderstorm and lightning phenomena, integrated with scenarios inspired by real-world weather events. As shown in Table 1, we summarize the lightweight parameter set used for the scenes presented in this section. All renders are produced using Houdini’s native volumetric rendering engine to ensure high-quality visual outputs.

Table 1. Overview of the lightweight parameter set used in the scenes presented in this section. DD: World dimensions (km); RR: Volume resolution; TmcsT_{\text{mcs}}: Computation time in seconds per frame(s).
Fig. Scene DD (km) RR TmcsT_{mcs} (s)
6(a) Single Cell 18×12×1818\times 12\times 18 450×301×450450\times 301\times 450 0.48
6(b) Multicell 20×15×2020\times 15\times 20 500×376×500500\times 376\times 500 0.81
6(c) Squall Line 30×15×3030\times 15\times 30 500×251×500500\times 251\times 500 0.54
6(d) Super Cell 30×15×3030\times 15\times 30 500×251×500500\times 251\times 500 0.68
7(a) Florida 11×12×1111\times 12\times 11 287×309×287287\times 309\times 287 0.24
7(b) New Mexico 11×13×1111\times 13\times 11 383×438×383383\times 438\times 383 0.59
7(c) Japan 11×13×1111\times 13\times 11 383×444×383383\times 444\times 383 0.64
7(d) California 10×7×1010\times 7\times 10 340×247×340340\times 247\times 340 0.28

6.1. Thunderstorms Variation

We simulate four common types of thunderstorms that occur within a mesoscale convective system (MCS), capturing their distinctive characteristics and dynamic behaviors:

  • Single Cell: Single cell thunderstorms, often referred to as “popcorn” convection, are small, isolated storms characterized by their brief lifespan, typically lasting less than an hour. These storms are visually compact with a single updraft and downdraft cycle, forming as isolated towering cumulus clouds.

  • Multicell: Multicell thunderstorms consist of clusters of individual convective cells, each at varying stages of development. They appear as a dynamic structure where new cells continuously form along the gust front, sustaining the system for several hours.

  • Squall Line: A squall line is a linear arrangement of thunderstorms, visually identifiable by a long and narrow band of cumulonimbus clouds. These storms can extend hundreds of miles in length but are typically narrow, often around 10-20 miles wide.

  • Supercell: The supercell is the most visually striking and organized thunderstorm type, dominated by a large, rotating updraft known as a mesocyclone. These storms feature massive, tilted, and rotating cloud structures, often rising up to 50,000 feet. The mesocyclone can span up to 10 miles in diameter and persists for hours.

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Figure 6. Comparison between simulation results (a-d) and realistic observations (e-h) for four storm types: single cell, multicell, squall line, and supercell. This layout highlights the structural similarities between our simulated storms and their real-world counterparts.

These simulated variations allow us to explore the diverse behaviors and impacts of thunderstorms within an MCS. The visual comparison of these thunderstorm types is presented in Figure 6, highlighting their structural differences and unique features.

6.2. Severe Weather Phenomena

To explore the impact of thunderstorms in real-world scenarios, we reference severe weather events in geographically diverse locations. These events illustrate the variability and intensity of thunderstorms in different environments:

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Figure 7. Simulation cloudscapes inspired by real-world weather events. (a) Florida Biscayne National Park, (b) New Mexico Chiricahua Mountains, (c) Japan Fuji-Hakone-Izu National Park, (d) California coastal atmospheric river. These visualizations highlight the geographical diversity and meteorological phenomena captured in our simulation framework.
  • Biscayne National Park, Florida (25.3692, -80.243): On March 13, 1993, a tropical storm occurred in this region during the spring, bringing heavy rainfall and strong winds.

  • Chiricahua Mountains, New Mexico (35.7943, -106.443): On July 14, 1989, a severe thunderstorm swept through this mountainous area during the summer, producing heavy rain, hail, and strong winds.

  • Fuji-Hakone-Izu National Park, Japan (37.745, -119.54): On January 10, 2011, a sudden winter ;p‘thunderstorm formed in this region, resulting in brief but intense snowfall.

  • Monterey Bay, California (36.6197, -121.906): On December 2, 2012, an atmospheric river brought heavy rainfall and thunderstorms along the California coast, accompanied by strong gusty winds.

These events serve as the basis for our simulations, showcasing the capability of our model to replicate the diverse and complex phenomena associated with severe weather. The corresponding visualizations are presented in Figure 7.

7. VALIDATION

We adopt the evaluation methods outlined in (Shen et al., 2020) and (Cesana et al., 2019), utilizing altitude-based quantitative data to analyze cloud behavior across different atmospheric layers. The results are presented in Figure 8, which illustrates the relationship between cloud fraction and storm height. Notably, the maximum cloud fraction corresponds to the storm’s peak altitude, consistent with the classical vertical distribution of cumulonimbus clouds in meteorology.

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Figure 8. Comparative visualization of cloud fractions under different storm systems. From left to right: single-cell, multi-cell, squall line, and supercell. Each panel highlights the cloud fraction structure typical of these systems.

For further validation, we specifically selected four regions frequently studied in thunderstorm meteorology (Uman, 2001; Harris and Carvalho, 2018). Following the methodology from (Herrera et al., 2021), we examine the temporal evolution of the thunderstorm described in section 6.2. The evolution of cloud coverage is compared with real-world weather data from public sources, showing dynamic changes in the distribution of cloud cover. This comparison helps assess the accuracy of the model in capturing cloud coverage patterns observed in nature.we present the maximum storm development height along with the corresponding cloud fraction, providing a detailed overview of the storm’s vertical structure. This information is used to validate the model’s ability to simulate the typical altitude and distribution of clouds during storm development.

In line with the evaluation method in (Formenton et al., 2013), we analyze the evolution of lightning flash rates to capture the electrical activity of thunderstorms. The flash rate increases as the storm develops and gradually declines to zero as the storm dissipates. This trend reflects the typical lifecycle of a thunderstorm and its charge dynamics. Additionally, the frequency of lightning occurrences is examined in relation to altitude.

Figure 9 provides a comparison across Florida, New Mexico, Japan, and California, illustrating the model’s ability to capture a wide range of climatic behaviors and atmospheric dynamics, and confirming its consistency with observed storm structures and lightning patterns.

8. CONCLUSION AND FUTURE WORK

We have developed a comprehensive framework for simulating atmospheric phenomena during thunderstorms, focusing on thundercloud development, dissipation, and lightning generation through cloud electrification processes. This framework integrates a Grabowski-style extended warm cloud microphysics scheme with hydrometeor electrification processes to ensure the consistent coupling of cloud formation and lightning dynamics. The model simulates key atmospheric quantities, including microphysical properties such as vapor, cloud water, ice, and precipitated rain and snow, as well as electrodynamic properties like static charge distribution and lightning flashes. Our framework reproduces various thundercloud types, including single-cell, multicell, squall line, supercell, and atmospheric river formations, and simulates lightning dynamics using the Dielectric Breakdown Model (DBM), capturing both CG and IC lightning. Validation against real-world weather data demonstrates the model’s accuracy in simulating cloud structure, temperature evolution, cloud coverage, relative humidity, and lightning flash rates, with results benchmarked against national weather services and observed lightning activity.

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Figure 9. Comparison of cloud coverage evolution (top row) ,cloud fraction (middle row) and lightning activity evolution(bottom row)across four distinct regions: Florida, New Mexico, Japan, and California. The figures showcase the distinct patterns of atmospheric dynamics influenced by regional environmental conditions.

Future work will focus on expanding the framework to incorporate advanced atmospheric concepts and support a broader range of thunderstorm and lightning phenomena. Enhancements to thunderstorm microphysics, such as more detailed electrification processes and turbulence modeling, will improve the realism of lightning generation. The framework will also be extended to simulate additional thunderstorm types, including Mesoscale Convective Complex (MCC), Mesoscale Convective Vortex (MCV), and derechos, broadening its coverage of mesoscale systems. Furthermore, new lightning types such as anvil crawlers, bolts from the blue, and sheet lightning will be introduced, requiring refined electric field and branching models. Finally, scalability and real-time performance will be improved by leveraging advanced numerical techniques and hardware optimizations, enabling more detailed and efficient simulations.

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