Simulating Snowfall and Snow Adhesion on Surfaces

Nitin Kanchinadam & Sophie Tsai

Outline

1. Motivation
2. Snow Formation
3. Snowfall Dynamics
4. Properties of Roads
5. Snow & Surface Interactions
6. Overview
7. Future Work

Motivation

Motivation

• Traffic Safety: Snow on roads makes roads slipperier, reducing vehicles’ traction with the ground, making it easier lose control and get in an accident
• Every year, 900 are killed in traffic during snowfall
• Every year, 76,000 are injured in traffic during snowfall
• Pedestrian Safety: Snowy sidewalks lead to more injuries as people are more likely to slip and fall

Motivation

• Supply Chain Disruption: Snow makes the movement of raw materials & finished products throughout the world significantly more challenging & time-consuming
• 11.46B tons of freight moved by truck in 2023
• Was 10.93B tons in 2022
• Climate Change: As the climate changes, the severity to which snow impacts certain areas will change, creating a need for new methods to deal with the aforementioned issues

Snow Formation

How Do Snowflakes Form: Criteria

• Sufficient Humidity: Atmosphere must be saturated with water → depends on both temperature and vapor pressure
• Condensation Nuclei: Small particles in the air (e.g. dust and soot) act as starting points for condensation
• Abundant Water: Enough water must be present so droplets can keep growing
• Sub-Freezing Temperatures: Air temperature must be below 0°C so water in the air freezes onto the nuclei
• Nuclei Shape: Must be of specific shapes to form basic ice crystals
• Nuclei are not necessary under -40°C

How Do Snowflakes Form?

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Snowflake Size and Shape

• highly dependent on the environment temperature and humidity

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Liquid Water Content (LWC)

• Describes the amount of water in the snow
• Volume ratio of liquid water in the snow to the total volume of snow

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Wet vs Dry Snow

Wet Snow

• Snowflakes pass into positive temperature air layer, causing melting to occur and thus an increase in liquid water content (LWC)
• Water fills gaps between ice particles, minimizing air between them

Dry Snow

• Occurs when surface air temperature is below freezing
• Composed of air and ice, small LWC
• Powdery structure

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Snowflake Model - Moeslund et al.

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• T.B. Moeslund, C.B. Madsen, M. Aagaard, and D. Lerche (2005)
• Focused on realistic appearance for computer graphics
• Model based on physics balancing complexity and visual result
• Macroscopic view
• Main characteristics to model: Size, density, and shape

Snowflake Model - Moeslund et al.

• Size

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�+ random number from uncertainty�

• Density

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C (proportionality constant) -

Dry Snow: 0.170 kg/m²

Wet Snow: 0.724 kg/m²

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D: diameter (meters)

T: temperature (°C)

Snowflake Model - Moeslund et al.

• Shape
• Combined triangular polygons in random manner
• Layers determine the snowflake’s size
• Number of polygons set the snowflake’s density
• Fixed number of polygons per layer
• Wet Snow: 10 polygons
• Dry Snow: 40 polygons

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Snowflake Model - Moeslund et al.

• Observe effect of temperature
• Snowflakes generally have a realistic “fluffy structure”
• Seems unnatural when a snowflakes is extremely close to the “camera”
• Can update the position of around 100,000 snowflakes per second, scaling linearly*
• Uses non-optimized code on a 1.8GHz PC with 256MB Ram

Snowflake Model - Cellular Automata

• Mesoscopic approach
• Discrete space: Space broken into grid of cells
• Discrete state: Each cell can be in one of a finite number of cells
• Local rules: Cell state at next time step determined by current state of itself and neighbors
• Reiter’s Model (2004)
• Hexagonal cells: potential locations for ice formation
• State s_t(z): amount of water stored in cell z at time t
• frozen: s_t(z) ≥ 1
• boundary: not frozen, but at least one neighbor is
• nonreceptive: neither frozen nor boundary

Snowflake Model - Reiter’s Model

• Start:
• Origin cell 0: s₀(0):= 1
• All other cells: s₀(0):= β,
• β: fixed background vapor level
• 2 intermediate variables:
• u_t(z): amount of water that participates in diffusion
• v_t(z): the amount that does not participate
• If a cell is receptive, u_t​(z):=0 and v_t​(z):=s_t​(z)
• If a cell is nonreceptive, u_t​(z):​=s_t(z) and v_t​(z):=0.

Snowflake Model - Reiter’s Model

• Follows 2 local update rules:
• Constant Addition, for any receptive cell z,

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• Diffusion, for any cell z,

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u_t(z): amount of water that participates in diffusion

v_t(z): the amount that does not participate

𝛾 : vapor addition

α: vapor diffusion

ū_t⁻​(z): average of u_t^−​ for the 6 nearest neighbors of cell z

Snowflake Model - Reiter’s Model

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• Varying α, β, 𝛾 allows generation of a variety of realistic snowflake formations
• Reiter Snowflake Simulation
• Produces 2D symmetric structures
• Original Reiter’s model primarily focuses on diffusion control
• Missing interface control

Snowflake Model - Phase Field Model

• Used for modeling interfaces and microstructures
• Continuum-based using a diffuse interface
• Governed by partial differential equations derived from a free energy functional
• Demange, Zapolsky, Patte, and Brunel (2017)
• Two coupled variables:
• Φ: phase field distinguishing between ice (+1) and vapour (−1) phases
• u: reduced water vapor supersaturation
• Initially, u₀ is homogeneous.
• Growth kinetics: 2 non-conservative phase field equations

Snowflake Model - Demange et al.

• ∂_t​ϕ: rate of change of the phase field ϕ over time at a specific location
• A(n)²: square of the surface tension anisotropy function

Snowflake Model - Demange et al.

• -f’(ϕ): derivative of double-well potential f(ϕ)
• λB(n)g′(ϕ)u: anisotropic thermodynamic driving force for ice growth due to supersaturated environment
• a . : ��describe the formation and propagation of the ice/vapor interface itself

Snowflake Model - Demange et al.

• ∂_t​u: rate of change of the reduced supersaturation u over time at a specific location
• : diffusion of water vapor through the environment
• : accounts for the conversion of vapor into ice

Snowflake Model - Demange et al.

• Formation of complex 3D snowflake formations in the Nakaya diagram
• Quantitatively supported simulated morphology dependence on temperature and humidity
• Limited ability to simulate snowflake growth at very small supersaturations

Snowfall Dynamics

Modeling Snowfall - Moeslund et al.

• Moeslund et al. (2005)
• Particle system for falling snow
• Snowflake movement cause by 4 forces:
• Gravity
• Buoyant Force
• Lift Force
• Drag Force
• Fluid dynamics for the wind field

Modeling Snowfall - Moeslund et al.

• Gravity
• Buoyant Force: Force representing updrift by surrounding air
• Direction is always opposite of gravity
• Relatively small; Ignored due to no visual impact
• Lift Force: Force contributing to circular and irregular motion of snowflakes
• Caused by snowflake's shape and air turbulence
• Model incorporates randomness in initial rotation and radius influenced by wind speed

Modeling Snowfall - Moeslund et al.

• Drag Force: Represents air resistance and causes the snowflake to be carried by the wind

m_snow: snowflake mass

g: gravitational acceleration

U_maximum: maximum vertical velocity due to air resistance

Dry Snow: [0.5m/s; 1.5m/s]

Wet Snow: [1m/s; 2m/s]

U_fluid: velocity of air relative to snowflake

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Modeling Snowfall - Moeslund et al.

• Wind Field: instant of fluid dynamics and can be described by Navier-Stokes
• For this model, air is incompressible, inviscid, and has a constant density of 1
• Navier-Stokes can be simplified to the incompressible Euler equations

∇·: divergence

u: vector field of the velocity

p: pressure

Modeling Snowfall - Moeslund et al.

• Convection Step
• Semi-Lagrangian method
• Projection Step
• Helmholtz-Hodge decomposition

P_a: point to find velocity for

P_d: departure point

u: divergence-free velocity field

u*: velocity field after convection

∇p: gradient of the pressure field

Modeling Snowfall - Moeslund et al.

• Influenced by realistic behavior
• Smaller discretizations of the scene led to more complex snowflake paths
• Velocity of wind has large effect on movement
• Intuitive control
• Simplified physics
• Computational efficiency

Surface Properties

Key Road Properties

Physical Properties

• Absorptivity - the surface’s ability to take in radiation
• Reflective Radiation - the surface’s ability to bounce incoming radiation away from the body
• Emissivity - the surface’s ability to give off radiation

Thermophysical Properties

• Heat Flux - the rate at which heat flows either into or out from a surface
• Thermal Conductivity - a measure of a surface’s ability to conduct heat

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Methods of Heat Transfer

• Conduction - heat transfer between two objects that are directly in contact with each other
• Convection - the movement of particles through a substance, transferring their heat from hotter areas to colder areas
• Radiation - energy that is transferred through either waves or particles

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Convection is the least relevant method of heat transfer to our situation

LWC & Adhesiveness

Wet Snow

• Snowflakes are “sticky” and easily adhere to and accumulate on most surfaces
• As LWC increases, capillary forces increases forming strong adhesive bonds, until LWC in snow surpasses ~20%

Dry Snow

• Low adhesion strength to surfaces
• Lack of free water to form capillary actions
• Many air gaps occur at surface result in �lower contact area

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Water Content of Surfaces

• Surfaces can be saturated with water as it seeps into the cracks & imperfections in the surface
• Heat flux varies with the amount of water held by the surfaces

Modeling Road Conditions

Combination of two 1D models

• Interactions between Soil, Biosphere, and Atmosphere (ISBA)
• Specific version is ISBA-DF, which only models soil-atmosphere interactions
• Assumes roads have a thin permeable layer at their tops, which can contain water, affecting heat flux
• CROCUS
• Detailed snow model for avalanche forecasting
• Simulates energy, mass, and stratigraphy of snow cover as a function of meteorological conditions

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Modeling Road Conditions

Road Surface Temperature Evolution

Snow & Surface Interactions

Particle Interactions - Interparticle Forces

• Interparticle forces can cause snow particles to adhere to each other and form larger aggregates as they fall
• Hydrogen Bonding
• Van der Waals

Hydrogen Bonding

• Intermolecular force
• Occurs between polar molecules
• Responsible for cohesion of water

δ+

δ+

δ+

δ-

Hydrogen Bonding

Van der Waals Forces

• Intermolecular force
• 30x weaker than Hydrogen Bonding
• Temporary changes in molecule polarity
• Temporary attraction between molecules
• Can lead to chains of temporary dipoles

Modeling Snow Adhesion - DEM Simulation

Discrete Element Method (DEM)

• Tracks the motion and interaction of individual snow particles
• Eidevåg, Abrahamsson, Eng, Rasmuson (2020)
• Interparticle forces incorporated to track individual particles, particle-particle interactions, and the formation and breakage of agglomerates
• JKR (Johnson-Kendall-Roberts) model to describe adhesive interactions
• Ice particles are spherical and dry to represent aged road snow

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Modeling Snow Adhesion - DEM Simulation

• Normal Model:
• JKR (Johnson-Kendall-Roberts) with higher Tabor numbers
• Pull-off force:�
• Critical stick velocity:

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W: work of adhesion

R*: effective radius of contact

K₁ ≈ 0.9355�ρ: particle density�R: particle radius�W: work of adhesion�E*: effective elastic modulus

Modeling Snow Adhesion - DEM Simulation

• Tangential Sliding Model:
• Critical sliding force:

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• Tangential Rolling Model:
• Constant directional torque (CDT) model
• Torque:

μ_f: sliding friction coefficient

F_n​: normal contact force

F_c: pull-off force

μ_r​: rolling friction coefficient k_n​: elastic stiffness Δγ/γ�δ_n​: normal overlap�ω_r​: relative rotation rate

SNTHERM (CRREL, 1991) - Intro

• 1D energy and mass balance model of a snowpack
• Divides modeling into layers (e.g. snow & soil)
• Each layer contains more granular nodes
• Each node keeps track of:
• Temperature (K)
• Node Thickness (m)
• Density (kg/m^3)
• Grain Size (m)

SNTHERM (CREEL, 1991) - Layers

• Layers are a combination of:
• Water vapor (v)
• Liquid water (l)
• Ice (i)
• Air (a)
• Dry soil solids (d)
• Variables
• Ф = Porosity
• θ = Volume Fraction
• ρ = Density

SNTHERM (CREEL, 1991) - Conservation Equations

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• Equations were used to conserve the following quantities:
• Overall Mass
• Water Phase Transitions
• Momentum
• Energy

SNTHERM (CREEL, 1991) - Limitations

• 1-Dimensional
• Not representative of the real world
• Undisturbed Snowpacks
• Only models natural phenomena
• Does not adapt well to sudden changes such as:
• Snow plowing
• Compaction

SnowModel (Liston & Elder, 2006) - Intro

• 3D spatially distributed snow-evolution model
• Combination of 4 submodels
• MicroMet - meteorological forcing conditions
• EnBal - surface energy exchanges
• SnowPack - snow depth & water-equivalent evolution
• SnowTran-3D - snow redistribution by wind

SnowModel (Liston & Elder, 2006) - MicroMet

• Takes in weather station data as input
• Air temperature, Relative humidity, Wind speed, Wind direction, Solar radiation, Longwave radiation, Surface pressure, Precipitation
• Outputs meteorological forcing distributions over the input area’s topography

SnowModel (Liston & Elder, 2006) - EnBal

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• Variables
• α = albedo
• Q_si = solar radiation reaching Earth’s surface
• Q_li = incoming longwave radiation
• Q_le = emitted longwave radiation
• Q_h = turbulent exchange of sensible heat
• Q_e = turbulent exchange of latent heat
• Q_c = conductive energy transport
• Q_m = total energy flux available for melting snow
• Melt energy defined to equal 0
• T_m = surface temp necessary to begin melting snow

SnowModel (Liston & Elder, 2006) - EnBal

Conductive Heat Transfer

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Portion of Expanded Surface Temperature Equation

SnowModel (Liston & Elder, 2006) - SnowPack

• Snow depth and water-equivalent evolution model
• Snow samples taken from 24 locations around Alaska
• Predictions made based on similarity with the taken samples

SnowModel (Liston & Elder, 2006) - SnowPack

• Snow depth and water-equivalent evolution model
• Snow samples taken from 24 locations around Alaska
• Predictions made based on similarity with the taken samples

SnowModel (Liston & Elder, 2006) - SnowPack

• Variables
• A = nx(n+1) Covariance Matrix
• x_i = ith variable being considered
• a_ij = Cov(x_i, x_j)
• v_i = avg(x_i)
• s_i = ith discriminate score

SnowModel (Liston & Elder, 2006) - Results

• Previous models were limited in their scope of terrain
• SnowModel models snow accumulation on vegetative surfaces

SnowModel (Liston & Elder, 2006) - Results

• Previous models were limited in their scope of terrain
• SnowModel models snow accumulation on vegetative surfaces
• Step towards better modeling of rural & suburban areas

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SnowModel (Liston & Elder, 2006) - Limitations

• Unable to deal with radiation spikes due to gaps in forest canopy
• Only modeled on fully vegetated terrains

Overview

Snow Formation & Snowfall

Snow Formation

• Moeslund et al. - Procedural Model
• Reiter’s Model - Hexagonal Cellular Automata
• Demange et al. - Phase Field Model

Snowfall

• Moeslund et al. - Simplified Fluid Dynamics

Snow & Surface Interactions

• SNTHERM - 1990s
• 1D snowpack energy & mass balance model
• Only models undisturbed snowpacks
• Can’t handle sudden changes
• SnowModel - mid-2000s
• Aggregates and builds upon 4 earlier models
• Expands those models to work in scenarios with vegetative surfaces
• Can expand to more types of surfaces in the future

Future Work

Future Work

• More work on modeling wet snow & phase transitions during snowfall
• Expanding upon SnowModel
• Snow accumulation & distribution on surfaces that aren’t just vegetation (e.g. rural & suburban areas)
• Expanding upon the DEM model
• Focus on modeling snow

References

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