JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 20, 219-232 (2004)

Modeling of Volcanic Clouds using CML*RYOICHI MIZUNO, YOSHINORI DOBASHI** AND TOMOYUKI NISHITA
Department of Complexity Science and Engineering
The University of Tokyo
Tokyo, Japan**
Department of Electronics and Information Engineering
Hokkaido University
Sapporo, Hokkaido, Japan

In this paper, the model of volcanic clouds for computer graphics using the Coupled
Map Lattice (CML) method is proposed. In this model, the Navier-Stokes equations areused, and the equations are solved by using the CML method that can be applied as anefficient fluid solver. Moreover, to generate desired shape of volcanic clouds, some parameters that allow intuitively control of the shape are provided. Hence, in this system,
the behavior of the volcanic clouds can be calculated in practical calculation time, andby only changing some parameters various shapes of the volcanic clouds can be generated. Therefore, photo-realistic images/animations of various shapes of volcanic clouds
can be created efficiently by the proposed approach.
Keywords: volcanic clouds, coupled map lattice, modeling, visualization, animation,
computational fluid dynamics, cellular automaton

1. INTRODUCTION
The modeling of volcanic clouds is useful for natural disaster simulations, entertainment (e.g. games, movies), etc. However, there is not much research on the modelingof volcanic clouds. Especially, in the field of computer graphics, this kind of research has
almost not been done so far. Although there are several commercial modeling products,
for example [33], that can generate volcanic cloud images, they can only obtain the motion of volcanic clouds according to the orbits of some particles that are set by professional users. Therefore, we propose a modeling method for volcanic clouds using the
Coupled Map Lattice (CML) method. The CML method [15, 16, 27-31] is one kind ofcell dynamics, which generates patterns. The advantages of using the CML method are:(1) it is easy to implement and (2) it has low computational cost. Moreover, to generate
desired shape of volcanic clouds, some parameters that allow intuitive control of the
shape are provided, as is a graphical user interface (GUI) for interactively setting of theparameters. The goals of this approach are: (1) the realistic behavior of volcanic clouds
can be represented efficiently; (2) the various shapes of volcanic clouds can be generated
by changing only several parameters.

Received January 31, 2003; accepted July 4, 2003.
Communicated by Ruay-Shiung Chang.
*
A preliminary version of this paper has been presented at 2002 International Computer Symposium, 2002.

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2. RELATED WORK
Although there is almost no research on modeling of volcanic clouds in the field ofcomputer graphics, there is much research on complex behavior of fluids such as smoke[5, 7, 21], clouds [1, 2, 9, 13, 18], water [4, 6, 8], flames [17, 20, 32], etc.
Kajiya and Herzen proposed a simulation method for cloud by solving the Navier-Stokes equations [13]. However, at that time (1984), they could do calculations on asmall number of voxels due to the lack of the computational ability. Therefore their
method can not represent a realistic cloud. Foster and Metaxas proposed a method thatcan generate realistic motion of turbulent smoke on relatively small number of voxels [7],but this method is stable only when the time step is very small and costs a lot of time forthe calculation. Stam introduced the semi-Lagrangian advection scheme to calculate theadvection term of the Navier-Stokes equations [21]. By using the semi-Lagrangian advection scheme, it is possible to calculate the advection term of the Navier-Stokes equations stably even if the time step is large. Fedkiw et al. provided a technique called vorticity confinement that is applied to Stam’s model [5]. The vorticity confinement canrepresent small-scale vortexes lost during the numerical calculation process (refer [22]for details). However, the methods provided by Stam and Fedkiw et al. were premised ona relatively small space such as inside a room. Therefore their methods cannot take somefactors which are related to height, like the variation of the atmospheric density with respect to height, into consideration. To overcome this difficulty, we introduced some parameters that are functions of height, although the fluid solver of our model is similar tothe solver proposed by Fedkiw et al.
The CML method was originally developed by Kaneko and is an extended methodof cellular automaton [15]. Yanagita and Kaneko proposed a method for the modelingand characterization of cloud dynamics using the CML method [31]. Miyazaki et al. extended their work and applied it to the modeling of clouds [18]. Their method can generate realistic clouds in a practical calculation time. However, the method provided by Miyazaki et al. is designed only to animate clouds, and cannot be applied to volcanic clouds
directly.
In the field of earth and planetary sciences, there are many papers about the dynamics and modeling of volcanic clouds. One of the significant researches is proposed byWoods [25]. He analyzed the dynamics of the vertical structure of volcanic clouds, andproposed a one-dimensional model of volcanic clouds. Woods’ model was followed by
Dobran and Neri [3], Woods and Bower [26], and Neri and Macedonio [19]. However,since the models are one-dimensional, it is hard to apply them to three-dimensional
simulations directly. Valentine and Wohletz proposed an axisymmetric two-dimensionalmodel of volcanic clouds [24]. Moreover, their model was followed by Ishimine andKoyaguchi [11], Ishimine [12], and Susuki [23]. Their works made great progress since
they made it possible to visualize the shapes of volcanic clouds. However, since theirmodels are still axisymmetrically two-dimensional, they are also hardly to be directly
applied to three-dimensional simulations. Although Suzuki suggested that the model
proposed in [23] could be extended to be three-dimensional, the computational cost is too
expensive.

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3. BASIC IDEA
There are many factors that decide the shape of volcanic clouds. Among these factors, the eruption magnitude, buoyancy, decrease in volcanic cloud density, and temperatures of the magma and volcanic clouds are the most important. Since the ascending
current due to the temperature of the magma can be considered to be the eruption velocity, and the buoyancy due to the temperature of the volcanic clouds can be considered tobe the buoyancy generated by the difference between the volcanic cloud density and theatmospheric density. Hence, the temperatures of the magma and the volcanic clouds canbe simplified to enhance simulation speed. Therefore, the proposed model is designed bytaking the following important factors that decide the shape of the volcanic clouds.
∑ Eruption magnitude: The eruption magnitude is decided by the initial velocity anddensity of volcanic clouds, and it depends on the scale of the volcanic clouds.
∑ Buoyancy: The buoyancy is generated by the difference between the volcanic clouddensity and the atmospheric density. The typical conically shaped clouds are generated
due to the buoyancy.
∑ Decreasing of the volcanic cloud density: The volcanic cloud density can be decreased due to the loss of pyroclasts (fragments of magma). The diversities of the volcanic cloud shapes due to the differences of the contents inside the clouds are decided
by the variety in the distribution of the loss of the pyroclasts.
Moreover, we make our model efficient and stable by using stable fluid solvers,
which include the CML method that is a qualitative and efficient solver, and thesemi-Lagrangian advection scheme that is a stable solver even if the time step is large.

4. MODEL
4.1 Evolution of Velocity Field
Since the atmospheric fluid and the volcanic clouds have small viscosity, and theeruption velocity of the volcanic clouds is less than the speed of sound, it is assumablethat the following non-viscosity Naivier-Stokes equations can be used to describe thetime evolution of the velocity field.
∇ ⋅ u = 0,(1)∂u
= -(u ◊ —)u - —p + f ,∂t(2)

where u is a velocity vector, p is the pressure, and f is an external force that is applied to
the velocity field. Eq. (1) means that the inflow and the outflow of a unit cell are balanced, and is called the “continuity equation”. This equation is a constraint which projects the velocity vector to the divergent free field. The first term of the right hand of Eq.
(2) means the advection of the velocity vector, and is called the “advection term”. The
second term means the variation of the velocity caused by the gradient of the pressure,and is called the “pressure term”. The third term means that the velocity is varied by the

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external force, and is called the “external force term”. In the proposed method, Eq. (1)and the pressure term of Eq. (2) are approximated by using the CML method. That is, thefollowing approximated Navier-Stokes equation is obtained.
∂u∂t= − ( u ⋅ ∇ ) u + η∇ ( ∇ ⋅ u ) + f ,(3)

where η is a positive constant which means the rate of diffusion, and is called the “diffusion coefficient”. Actually, the diffusion coefficient controls the scale of vortexes.
The approximated Navier-Stokes equation does not need iterative calculation tosolve the continuity and the pressure effect, although iterative calculation is generally
needed to solve the Poisson equation in other methods.
4.2 Evolution of Volcanic Clouds
Volcanic clouds are transported by the atmospheric fluid, and the volcanic clouddensity is decreased due to the loss of the pyroclasts. Hence, the following equation forvolcanic cloud density ρ can be defined.
∂ρ∂t= − (u ⋅ ∇ ) ρ − κ ( z ) ρ ,(4)

where κ ( z ) is called the “decreasing rate” and should be set with the following two
considerations.
∑ Near the vent, the volcanic clouds include many large pyroclasts called the “volcanic
blocks”. Therefore, ρ decreases rapidly due to the fall of the volcanic blocks. To simulate this phenomenon, κ ( z ) needs to be set large in this region.
∑ In a higher region, the volcanic clouds consist of many small pyroclasts called “volcanic ash” and air, and then the pyroclasts are lost slowly. Therefore, κ ( z ) needs tobe set small in this region.

The diversity of the volcanic cloud shapes due to the differences of the constituentsof the volcanic clouds can be represented by setting κ ( z ) .
4.3 Buoyancy

Buoyancy occurs due to the difference between the volcanic cloud density and theatmospheric density, and affects the velocity field. The buoyancy fbuoy is defined by Eq.
(5). When the vertical component of fbuoy becomes negative, buoyancy works verticallydownward. In this case, fbuoy works as gravity.
fbuoy = α (ρatm(z) − ρ)z,(5)

where α is a positive constant which controls the strength of the buoyancy, z is a vertically upward unit vector, and ρatm(z) is the atmospheric density, which is defined as an
exponential function of height as:

MODELING OF VOLCANIC CLOUDS USING CMLρ atm ( z ) = ρ 0 exp  −z , He 

223(6)

where ρ0 is the atmospheric density at the sea level (z = 0), and He is the degree of variation of atmospheric density with respect to height, and called the “scale height”.
Buoyancy plays an important role in deciding the shape of the volcanic clouds, andthe following dynamics generate the conically shaped clouds.
∑ In the region where height is low is called the “gas thrust region”, the atmospheric density is less than the volcanic cloud density. Thus, the perpendicular component of fbuoy
becomes negative, and the buoyancy works perpendicularly downward. However, themomentum of the eruption is more dominant than buoyancy. Therefore, volcanic
clouds are delivered upwards.
∑ In a higher region which is called the “convective region”, the atmospheric density islarger than the volcanic cloud density. Thus, the perpendicular component of fbuoy becomes positive, and the buoyancy works perpendicularly upward. Hence, the volcanic
clouds are delivered upwards.
∑ The region which is higher than the convective region is called the “umbrella region”,
where the atmospheric density and the volcanic cloud density are almost balanced.
Therefore, fbuoy becomes almost 0, and the volcanic clouds are no longer delivered upwards.

The proposed approach can simulate realistic volcanic cloud behavior by satisfying
these dynamics.

5. FLUID SOLVER
5.1 Setting of Analysis SpaceThe analysis space is represented as nx × ny × nz voxels. Each voxel is a cube with auniform volume. The velocity vector u and the volcanic cloud density ρ are defined as
the state variables at the center of each voxel. In the initial state, u is set to a small value
randomly, and ρ is set to zero. However, for the voxels located in the mountain (thehatched squares in Fig. 1), u is set to a zero vector. Then, the decreasing rate κ ( z ) andthe strength of the side wind fwind(z) can be defined by dragging the corresponding control points of the Bézier curves (the dots in Fig. 2) of a GUI shown in Fig. 2. The horizontal direction of fwind(z) can be changed. The atmospheric density ρatm(z) is defined as
Eq. (6), and is shown in the GUI. The horizontal and vertical axes of the GUI are thestandardized value of each function and height, respectively. In the GUI, curve-a denotes
a decreasing rate, curve-b denotes the atmospheric density, and curve-c denotes thestrength of the side wind. Finally, the eruption velocity usrc and the initial volcanic clouddensity ρsrc that decide the eruption magnitude are assigned to the voxels corresponding
to the vent (the circle in Fig. 1). It is also possible to make the volcanic clouds erupt witha spread. In our method, by changing only these parameters, to represent the diversity ofthe volcanic cloud shapes can be represented.

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Fig. 1. Outline of the analysis space.

Fig. 2. The GUI for setting the functions ofheight.

5.2 Sequential Solver
The time evolution of the volcanic cloud behavior can be obtained by iterating asfollows. In these processes the velocity vector u and the volcanic cloud density ρ areupdated.
1. Add force: “Add force” is the process of adding an external force to the velocity fluid.
2. Advect: “Advect” is the process to advect the state variables.
3. Pattern: “Pattern” is the process of generating vortex patterns using the CML method.
4. Decrease: “Decrease” is the process of decreasing the volcanic cloud density withrespect to the loss of the pyroclasts.
5.2.1 Add force
In the Add force process, the effect of the external force as the third term of the righthand side of Eq. (3) is calculated. In our method, the external force f is the sum of thebuoyancy and the strength of the side wind fwind(z). Hence, by assuming that f is unchangeable within a time step ∆t , the equation for updating the velocity vector u is:

u*1 = u + (fbuoy + f wind ( z ))∆t ,(7)

where u*1 is the updated velocity vector in this process.
5.2.2 Advect

In the Advect process, the effects of the advection as the first term of the right handof Eq. (3) and the first term of the right hand of Eq. (4) are calculated. For this calcula-MODELING OF VOLCANIC CLOUDS USING CML

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tion, we use the semi-Lagrangian advection scheme. As illustrated in Fig. 3, a path p(x, s)
is defined as a parametric function of time parameter s by tracing a particle, which islocated in position x backwards along the velocity field at time t. Thus, p(x, s) represents
the position where the particle existed at time (t − s).

Fig. 3. Path setting in the semi-Lagrangian advection scheme.

The state variables at position x at time (t + ∆t) are advected from position p(x, ∆t).
Hence, the equations for updating the state variables are:u*2 (x) = u*1 (p(x, ∆t )) ,(8)

ρ *1 (x) = ρ (p(x, ∆t )) ,(9)

where u*2 and ρ *1 are the updated velocity vector and the updated volcanic clouddensity in this process, respectively.
5.2.3 Pattern

In the Pattern process, the effect of the generating vortex patterns as the second termof the right hand of Eq. (3) is calculated. The CML method is used for this calculation.
The equation for updating the velocity vector u*2 is:u* = u*2 + η∇(∇ ⋅ u)∆t ,(10)

where u* is the updated velocity vector in this process, and also the final updated velocity
vector in the all sequential processes. A discrete version of Eq. (10) is shown in Eq. (11).
Here, we only describe the equation for updating ui,j,k, which is the u component of thevelocity vector u(= (u, v, w)) of voxel (i, j, k).
u*i,j,k = u*i,j,k + h{(ui+1,j,k + ui-1,j,k - 2ui+1,j,k) + (vi+1,j+1,k - vi+1,j-1,k - vi-1,j+1,k + vi-1,j-1,k +wi+1,j,k+1 - wi+1,j,k-1 - wi-1,j,k+1 + wi-1,j,k-1)/4}Dt,(11)
where u*i , j ,k is the u component of the velocity vector of the voxel (i, j, k) after being
updated by the Pattern process. The v and w components can be updated similarly.
5.2.4 Decrease

In the Decrease process, the decrease in volcanic cloud density as the second termof the right hand of Eq. (4) is calculated. The equation for updating the volcanic cloud

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density ρ *1 is:ρ * = ρ *1 − κ ( z ) ρ∆t ,(12)where ρ* is the updated volcanic cloud density in this process, and also the final volcanic
cloud density in the all processes.
u* and ρ* are the velocity vector and the volcanic cloud density after updated while
a time step ∆t, respectively.

6. RESULTS
The images generated by the proposed method are shown in Figs. 4-7. Figs. 4 and 5
show the volcanic clouds when there is no side wind. The relative profiles of the heightfunction are also shown in Figs. 4-6. Fig. 4 shows the case when the decreasing rate in
the gas thrust region is set to a relatively large, so that the volcanic clouds include manylarge pyroclasts. Conical volcanic clouds are generated as a result. Fig. 5 shows the casewhen the decreasing rate in the gas thrust region is set to be relatively small, hence thevolcanic clouds consist of small pyroclasts and air. Spreading volcanic clouds are generated as a result. Fig. 6 shows a sequence of images of an animation of the volcanic clouds
affected by side wind. The parameters except the strength of the side wind are set as inFig. 4. The sharpness of the conic shape of the volcanic clouds depends on the decreasing
coefficient in the gas thrust region (curves-a in Figs. 4-6), and the curved condition of the
volcanic clouds can be controlled by adjusting the strength of the side wind (curve-c in
Fig. 6).

Fig. 4. Conical volcanic clouds generated by the proposed method. The diffusion coefficient(curve-a) when z is small was set relatively large.

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Fig. 5. Spreading volcanic clouds generated by the proposed method. The diffusion coefficient(curve-a) when z is small was set relatively small.

All of the images are visualized as the simulation results on 100 × 100 × 100 voxels,
by using the rendering method proposed by Dobashi et al. [14]. The computational timeof the simulation was approximately 1 second per time step on an Intel Pentium 4 2.8
GHz CPU machine, and the computational time was almost proportion to the number ofvoxels. When the CML method is not used, the simulation time is several seconds at least
depending on the Poisson equation solver.
Fig. 7 shows photographs of real volcanic clouds and the images of volcanic clouds
generated by the proposed method for comparison. The shapes of clouds’ outline of thephotographs are similar to the images generated by the proposed method. This can show
the representation ability of our model.

7. CONCLUSIONSIn this paper, a method for modeling volcanic clouds using the CML method is presented. The major features of our method are:
∑ The realistic behavior of volcanic clouds is represented by considering the eruption
magnitude decided by the eruption velocity and initial volcanic cloud density, thebuoyancy generated by the difference between the volcanic cloud density and the atmospheric density, and the decreasing of the volcanic cloud density due to the loss ofthe pyroclasts.
∑ Various shapes of volcanic clouds can be generated by changing only some parameters.
∑ An efficient simulation is achieved by using the CML method.

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1500th frame

The side wind (curve-c) is added.

250th frame

500th frame

750th frame

1000th frame

1250th frame

1500th frameFig. 6. Volcanic clouds affected by a side wind.

MODELING OF VOLCANIC CLOUDS USING CMLImages generated by the proposed method.

229

Photographs.

Fig. 7. Comparison photographs of real volcanic clouds and volcanic clouds generated by the
proposed method.

To enhance the reality of the images generated by our method, we should take account the expansion of the volcanic clouds due to the mixing of the pyrocalsts and thesurrounding air into the model. Moreover, to reduce the calculation time of the simulation, it is a good idea to implement the proposed model by using graphics hardware. Harris et al. proposed a fast calculation method of CML using graphics hardware [10], although graphics hardware is generally used for rendering. The implementation of the
CML part of the proposed model using graphics hardware will be a part of future work.

ACKNOWLEDGMENT
We thank Bing-Yu Chen (National Taiwan University) for many useful suggestions.

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Ryoichi Mizuno was born on July 31, 1978 in Tokyo, Japan.
He received his Bachelor of Science and Engineering from
Sophia University in 2001, and received his Master of Science in2003 from The University of Tokyo. Now, he is pursuing Ph.D.
degree of The University of Tokyo. His research interests centerin computer graphics including fluid models.

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Yoshinori Dobashi was born on July 14, 1969 in Hiroshima,
Japan. He received his BE, ME and Ph.D. in Engineering in 1992,
1994, and 1997, respectively, from Hiroshima University. He
worked at Hiroshima City University from 1997 to 2000 as a research associate. Now, he is an associate professor at Hokkaido
University in the graduate school of engineering, Japan since
2000. His research interests center in computer graphics including
lighting models.

Tomoyuki Nishita is a professor in the Department ofComplexity Science and Engineering (also in the Department ofInformation Science) at The University of Tokyo, Japan since1988. He taught at Fukuyama University from 1979 to 1998. He
was an associate researcher in the Engineering Computer Graphics Laboratory at Brigham Young University from 1988 to 1989.
His research interests center in computer graphics including
lighting models, hidden-surface removal, and antialiasing. Nishita
received his BE, ME and Ph.D. in Engineering in 1971, 1973, and1985, respectively, from Hiroshima University.