Simulation of Snowflakes By Using Cellular Automaton Model

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Simulation of Snowflakes By Using Cellular
Automaton Model

• Sükriye ÖZ
• Turkish State Meteorological Service, Ankara,
  Turkey
• soz_at_meteor.gov.tr
• Prof. Dr. Bülent KUTLU Gazi University
  Institute of Science and Technology, Ankara,
  Turkey
• bkutlu_at_gazi.edu.tr

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• A snowflakes is a letter to us from the sky
• A diamond is a letter from the depth
• Iwanami 1937

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• Snowflakes formation in the atmosphere conditions
• Snowflakes formation model examples
• (survey of cellular automaton )
• Cellular Automaton Algorithm for Lattice Gas
  Model
• Conclusion and Recommendations

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1.Introduction

• A Knowledge of cloud ice particle behavior is of
  great importance for theoretical and applied
  cloud physics. Ice crystals in clouds in the
  atmosphere have shapes, which relate to their
  density, terminal fall velocity, growth rate and
  radiative properties. In calculations for climate
  change predictions, forecasting of precipitation
  and remote sensing retrievals, idealized crystal
  shapes such as columns, needles, plates and
  dendrites are often assumed. Causal observation
  shows that snow particles falling from the sky
  appear in a great variety of shapes Korolev
  G.,2000. In the atmosphere above us, this water
  vapor condenses directly to form solid ice.

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• The process begins with some nucleus, typically a
  tiny dust, aerosol exp. grain, to which water
  molecules can easily attach. If the humidity of
  the air is above 100 percent (the air is then
  said to be supersaturated), water molecules
  freeze onto the dust nucleus, forming a tiny
  piece of ice, which subsequently grows into a
  snow crystal as more water molecules condense out
  of the air Figure 1.

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• In 1951 the International Commission on Snow and
  Ice produced a fairly simple and widely used
  classification system for solid precipitation
  ICSI, 2003. This system defines seven principal
  snow crystal types as plates, stellar crystals,
  columns, needles, spatial dendrites, capped
  columns, and irregular forms Figures

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Examples of several diffrent morphological types
of snow crystals
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• In the atmosphere above us, this water vapor
  condenses directly to form solid ice. The process
  begins with some nucleus, typically a tiny dust,
  aerosol exp. grain, to which water molecules can
  easily attach. If the humidity of the air is
  above 100 percent (the air is then said to be
  supersaturated), water molecules freeze onto the
  dust nucleus, forming a tiny piece of ice, which
  subsequently grows into a snow crystal as more
  water molecules condense out of the air

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• In 1951 the International Commission on Snow and
  Ice produced a fairly simple and widely used
  classification system for solid precipitation
  ICSI, 2003. This system defines seven principal
  snow crystal types as plates, stellar crystals,
  columns, needles, spatial dendrites, capped
  columns, and irregular forms

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• By growing snow crystals in the laboratory under
  controlled conditions, one finds that their
  shapes depend on the temperature and humidity.
  This behavior is summarized in the morphology
  diagram, shown at right, which gives the
  crystal shape under different conditions

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Figure 3. Dentritc snowflake growth at
Laboratory Libbrecth K. G. 2000.
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• Recently varieties of models have been used for
  simulation of dentritic growing of snowflake.
  Clouds, thunderstorms, galaxies and, fluids are
  self-developing systems. In addition to their
  behavior molecules, physical systems and organic
  systems are simulated with statistic mechanic
  such as complex patterns that enables of
  understanding cellular automation modeling.
  Firstly, It had been developed as a one
  dimensional snowflakes growing modelling with
  cellular automata by S. Wolfram Wolfram S.,
  1083.

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Realtime Cloud Simulation and Rendering Noah
Brickman UC Santa Cruz David Olsen UC Santa
Cruz Gillian Smith UC Santa Cruz They use
cellular automata to model cloud dynamics,
providing realistic looking clouds at realtime
rates. However, unlike previous physically based
simulations, we also offer an ability to control
the shape and appearance of clouds through custom
shaping routines. They were able to create and
animate four different types of clouds examples
are shown in Figures 1 and 2. These models
include Cumulonimbus capillatus incus,
Cumulus humilis, Altocumulus castellanus and
Cirrus. The initial number of cells was set at an
array of 256x256x20. The only parameter set was a
pre-set call to a particular cloud distribution
function.
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An example of a slightly less common type of
cloud the Cirrus. It is generally not produced
by other systems. Due to the wispiness and
complexity of its shape
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Cumulonimbus Capillatus Incus cloud rendered
using our system. This is the most common kind of
cloud seen in other cloud rendering systems.
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• Lattice Gas Modeling is tried by Packard at
  cellular automata Packard N. H, 1986. This
  method provided that different crystal simulation
  and patterns are explained Ndkarni G., 1995.
  Two dimensional cellular automata modeling was
  performed by Packard N. H. and Wolfram S.
  Packard N. H. and Wolfram S.,1995 .
  Furthermore Suzudo worked on crystallization
  with two dimensional Cellular automata Suzudo
  T., 1998.

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Comparison of a natural crystal (left) and a
Packard CA (right)
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• Howewer Mean Field Kinetic Equations Modeling
  super-saturated field (chemical potential) is
  used to obtain which is begins one seed and is
  continue unstable for four branching
  snowflakes simulation. Gouyet J. F., 1999. The
  same model is utilized total energy to minimize
  temperature boundary conditions in two
  dimensional square lattice in order to get also
  many branching snowflakes simulation Vera F.
  2002.

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Fourfold "snowflake" grown on a 200200 square
lattice at T 0.1Tc. The vertical axis shows
the concentration a seed (p1, radius R10) has
been initially put in the sample, surrounded
a concentration fixed on a large circle at p
0.02.
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• It is from the physicists to develop cellular
  automata alternative to differential equations in
  modeling thermodynamic and that is how
  statistical physics began. This is resulted in
  investigation of cellular automata models for
  physical system with an emphasis on spin systems
  models for various for of random based on two
  dimensional models such as ferromagnetism
  according Ising Model Creutz M.,1986,. This
  model 1925 represents the material as a network
  in which is magnetic state. This state in the
  case one of the two orientations of the spins of
  the certain electrons depending on the state of
  the neighboring nodes Kutlu B. and N. Aktekin,
  94 Kutlu B. and N.Aktekin, 1995 Kutlu B.,
  1997.

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• In this study, the formation and growth of
  snowflakes are examined by using a Cellular
  Automaton Lattice Gas Model. For this purpose,
  the algorithm of Cellular Automaton, developed by
  Creutz, Creutz, 1986, is adapted to the
  Lattice Gas Model for understanding the magnetic
  phase transition problems with spin -1/2 Ising
  Model to have the simulation of snowflakes as in
  the atmosphere. It is well known that, the
  particular density and the environment
  temperature are higly effective in the formation
  and variations of snowflakes.

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Cellular Automaton Algorithm for Lattice Gas
Model
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• In this model each cell on the square lattice
  corresponds to three variables and the value to
  be assumed by the variables in a cell and the
  values of the neighbor variables will be
  determined according to the following rule. The
  first variable in this model ni is the status of
  particle and its value may be 0 or 1 thus ni
  0 if the site is empty, 1 if it is full.
  Lattice gas energy HI can be defined as follows

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• The first sum being only over nearest-neighbour
  pairs of sites on the lattice. The e is the
  interaction energy between particles centered on
  sites and m is the effective chemical potential.
  The second variable is for the momentum variable
  (the demon) conjugate to the particles. The
  kinetic energy associated with the demon, Hk , is
  an integer, which is equal to the change ,in the
  lattice gas energy for any flip (01 or 10)
  which also lies in the interval (0,4). The total
  energy
• HHIHK
• is concerved. The third variable provides a
  checkerboard style updating, and so it allows the
  simulation of the ising model on a cellular
  automaton.

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• The black sites of the checkboard are updated and
  then their colour is changed into white white
  sites are changed into black without being
  uptated. The uptading rules for the spin and the
  momentum variables are as fallows for a site to
  be uptated its spin is flipped and the change in
  the Ising Energy (internal energy), dH1, is
  calculated. If this energy change is transferable
  to or from the momentum variable associated with
  this site, such that the total energy, H1, is
  calculated. If this energy change is the
  transferable to or from the momentum variable
  associatedwith this site, such that the total
  energy H is concerved, then this changed is done
  and the momentum is appropiriately changed
  otherwise the spin and the momentum are not
  changed.

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• Lattice Gas Cellular Automation Algorithm was
  arranged in the following in order to realize the
  simulation experiment of the snow particle. A
  nucleus was placed at the center of 80x80 square
  cell and it was kept fixed during the whole
  simulation process. For the values of the
  chemical potential µ 0, µ 1, µ 2, µ 3, the
  side cells of the lattice were considered to be
  full, initially.

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• It means to take all the effects into account.
  The simulation of snowflakes are realized by
  using a dynamical algorithm which allows to
  examine the formation of particles in the
  chemical potential (µ) and nearest neighbors
  interaction (J). This algorithm is run at
  diffrent lattice temperatures for determining the
  particles around nucleus and how to organize for
  J-1 and µ0,1,2,3 on a square lattice, L8080
  in a space with constant chemical potential (µ).
  The spatial variations, for the probability of
  particles occupying a cell on the lattice were
  calculated and using these variations and the
  forms of snowflakes were obtained as a result of
  the simulations. Depending on the lateral
  conditions for µ3, starred and dentritic forms
  are obtained and it was seen that the spatial
  chemical potential are highly effective on the
  organization of particles

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Result of simulation
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Recommendations
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• Dynamic formation of the snowflakes was examined
  and a computer-based simulation was developed by
  using the Cellular Automation Lattice Gas Model.
  In this study, atmospheric conditions under which
  snowflake forms was represented by µ spatial
  chemical potential, mean kinetic energy which is
  a measure of the environmental temperature and
  the particle density in the environment. It was
  observed that the simulation made with this
  algorithm produces results that are in harmony
  with the snowflake shapes existing in nature, for
  the µ 3 of the spatial chemical potential. This
  study was performed on two-dimensioned lattice.

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• When formations for µ 3 is examined,
  embranchments are observed along the diagonal
  shapes. When it was compared with the snowflakes
  which exist in the nature, the reason why the
  formations had four branches, instead of six
  ones, is understood better because the study was
  performed on two-dimensioned lattice for better
  interactions with the closest neighbor. In order
  to obtain hexagonal or six-branch snowflakes, the
  algorithm needs to be established on the triangle
  or hexagonal lattice structure. The findings
  reveal that cellular automation lattice gas model
  is an alternative method in the simulation of the
  snowflake formation,

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• in comparison to the other methods. In addition
  to using it to obtain the formation of snowflakes
  in the nature, the cellular automation model can
  also is used for the simulation of many physical
  events including macro-scale particle
  organization. This model can be used not only for
  obtaining natural shapes of snowflakes but also
  for simulating many other physical events i.e.
  cloud and thunderstorm formation which comprise
  macro-scale particle organizations Macke A.,
  1997 Brickman N., 2003 Iudin D.I., 2003.

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