cloud kwang hee ko september, 27, 2012 this material has been prepared by y. w. seo
TRANSCRIPT
Cloud
Kwang Hee KoSeptember, 27, 2012
This material has been prepared by Y. W. Seo.
Introduction Clouds are a ubiquitous feature of our
world Provide a fascinating dynamic backdrop,
creating an endless array of formations and patterns.
Integral factor in the behavior of Earth’s weather systems
Important area of study for meteorologists, physicists, and even artists
Introduction Clouds play an important role when making
images for flight simulators or outdoor scenes. Clouds’ color and shapes change depending on
the position of the sun and the observer.
The density distribution of clouds should be defined in three-dimensional space to create realistic images.
Introduction The complexity of cloud formation,
dynamics, and light interaction makes cloud simulation and rendering difficult in real time. Ideally, simulated clouds would grow and
disperse as real clouds do.
Simulated clouds should be realistically illuminated by direct sunlight, internal scattering, and reflections.
Cloud Dynamics Simulation Clouds are the visible manifestation of
complex and invisible atmospheric processes. Fluid dynamics governs the motion of the air, and
as a result, of clouds. Clouds are composed of small particles of liquid
water carried by currents in the air. The balance of evaporation and condensation is
called water continuity. The convective currents are caused by
temperature variations in the atmosphere, and can be described using thermodynamics.
Cloud Dynamics Simulation Fluid dynamics, thermodynamics, and
water continuity are the major processes. The physics of clouds are complex.
By breaking them down into simple components, accurate models are achievable.
Cloud Radiometry Simulation Clouds absorb very little light energy. Instead, each water droplet reflects, or
scatters nearly all incident light. Clouds are composed of millions of these tiny
water droplets. The light exiting the cloud reaches your eyes,
and is therefore responsible for the cloud’s appearance.
Cloud Radiometry Simulation Accurate generation of images of clouds
requires simulation of the multiple light scattering.
The complexity of the scattering makes exhaustive simulation impossible .
Instead, approximations must be used to reduce the cost of the simulation.
Efficient Cloud Rendering After efficiently computing the dynamics
and illumination of clouds, there remains the task of generating a cloud image. A volumetric representation must be used to
capture the variations in density within the cloud.
Rendering such volumetric models requires much computation at each pixel of the image.
The rendering computation can result in excessive rendering times for each frame.
Efficient Cloud Rendering The concept of dynamically-generated
impostors A dynamically-generated impostor is an image
of an object. The image is generated at a given viewpoint,
and then rendered in place of the object. The result is that the cost of rendering the
image is spread over many fames. Useful for accelerating cloud rendering
Physically-based Simulation on GPUs Using the GPU for simulation does more than just
free the CPU for other computations. It results in an overall faster simulation.
GPU implementations of a variety of physically-based simulations outperform implementations.
General-purpose computation on GPUs has recently become an active research area in computer graphics.
Cloud Dynamics The dynamics of cloud formation, growth,
motion and dissipation are complex.
To understand the dynamics is important.
To choose good approximations allows efficient implementation.
The Equations of Motion Assume that air in the atmosphere is and
incompressible, homogeneous fluid. Incompressible if the volume of any sub-
region of the fluid is constant over time. Homogeneous if its density is constant in
space. These assumptions do not decrease the
applicability of the resulting mathematics to the simulation of clouds.
The Equations of Motion The motion of air in the atmosphere can
be described by the incompressible Euler equations of fluid motion
where ρ is the density of the fluid. B is buoyant acceleration, and f is acceleration due to other forces.
Parcels and Potential Temperature A conceptual tool used in the study of
atmospheric dynamics is the air parcel. The parcel approximation is useful in
developing the mathematics. When a parcel changes altitude without a
change in heat, it is said to move adiabatically.
We can account for adiabatic changes of temperature.
Parcels and Potential Temperature The potential temperature, Θ, of a parcel of
air can be defined as the final temperature
∏ is called the Exner function, Rd is the gas constant.
Buoyant Force Change in the density of a parcel of air
relative to its surroundings result in a buoyant force on the parcel.
If the parcel’s density is less than the surrounding air, this force will be upward.
If the parcel’s density is greater, the buoyant force will be downward.
The density of an ideal gas is related to its temperature and pressure.
Buoyant Force A common simplification in cloud modeling
is to regard the effects of local pressure changes on density as negligible
where g is the acceleration due to gravity and qH is the mass mixing ratio of hydrometeors.
Environmental Lapse Rate The Earth’s atmosphere is in static equilibrium. The hydrostatic balance of the opposing forces
of gravity and air pressure results in an exponential decrease of pressure with altitude
Here, z is altitude, and P0 and T0 are the pressure and temperature at the base altitude.
Saturation Mixing Ratio Cloud water continuously changes from
liquid to vapor and vice versa. The water vapor mixing ratio at saturation
is called the saturation mixing ratio, denoted by qVS(T,p)
with T in Celsius and p in Pa.
Environmental Lapse Rate The water mixing ratios at a given location
are affected both by advection and by phase changes.
The rates of evaporation and condensation must be balanced, resulting in the water continuity equation
Where C is the rate of condensation.
Thermodynamic Equation The potential temperature of saturated air cannot be
assumed to be constant. If latent heating and cooling due to condensation
and evaporation are the only non-adiabatic heat sources, then the first law of thermodynamics results in
where L is the latent heat of vaporization of water.
Vorticity Confinement Vorticity confinement works by first
computing the vorticity , from which a normalized vorticity vector field
is computed. From these vectors we can compute a
force that can be used to replace dissipated vorticity back in
Vorticity Concept In fluid dynamics, the vorticity is a vector that
describes the local spinning motion of a fluid near some point, as would be seen by an observer located at that point and traveling along with the fluid.
One way to visualize vorticity is this: consider a fluid flowing. Imagine that some tiny part of the fluid is instantaneously rendered solid, and the rest of the flow removed. If that tiny new solid particle would be rotating, rather than just moving with the flow, then there is vorticity in the flow.
From wikipedia
Solving the Equations Fluid Flow
Water Continuity
Thermodynamics
Solving the Equations (Fluid Flow) The cloud model is based on the equations of
fluid flow. The simulator is built on top of a standard fluid
simulator. Solve the equations of motion using the stable
two step technique described . First, use the semi-Lagrangian advection technique Second, the intermediate field is made.
incompressible using a projection method based on the Helmholtz-Hodge decomposition .
Solving the Equations (Fluid Flow) The projection is performed by solving for
the pressure using the Poisson equation
with pure Neumann boundary conditions
Subtract the pressure gradient from u’
Solving the Equations (Water Continuity) The changes in qV and qC are governed by
advection of the quantities as well as by the amount of condensation and evaporation.
Solve equations in two steps First, advect each using the semi-Lagrangian
technique mentioned. Second, at each cell, compute the new mixing
ratio as follows
Solving the Equations (Thermodynamics) Potential temperature is advected by the
velocity field. The temperature increases by an amount
proportional to the amount of condensation, and is able to update it as follows.
Implementation Solve the equations on a grid of voxels. Use a staggered grid discretization of the
velocity and pressure equation. This means that pressure, temperature, and
water content are defined at the center of voxels.
This method reduces numerical dissipation. It prevents possible pressure oscillations that
can arise with collocated grids.
Interactive Applications Cloud simulation is a very computationally
intensive process. It is usually done offline.
Simulations of phenomena such as clouds have the potential to provide rich dynamic content for interactive applications.
Interactive Applications Integrate the cloud simulation into
SkyWorks cloud rendering engine. “Simulation of Cloud Dynamics on Graphics
Hardware” SkyWorks was designed to render scenes full of
static cloud very fast. It recomputes the illumination of the clouds,
and then uses this illumination to render the clouds at runtime.
Cloud Rendering Convert the simulation’s current cloud
water texture into a true 3D texture, which is then used to render the cloud for multiple frames. Rendering directly from the flat 3D texture is
too expensive. The conversion is overall much faster.
A simulation time step dose not complete every frame.
The generation of the 3D texture is included in the simulation amortization.
It doesn’t affect our interactive frame rates.
Cloud Illumination To create realistic images of clouds, we
must account for the complex nature of their interaction with light. Light has been scattered many times by the
tiny water droplets in the cloud. This is what gives clouds their soft, diffuse
appearance. A full simulation of multiple scattering requires
the solution of a double-integral equation.
Cloud Illumination A full simulation of multiple scattering
requires the solution of a double-integral equation.
Cloud water droplets scatter most strongly in the direction of travel of the incident light, or forward direction.
Example