The study of clouds has never been limited to scientists alone. Artists, poets, and even daydreamers watch with interest as the clouds constantly change before their eyes. At any one time, nearly half of the earth's surface may be obscured by cloud cover [Wallace and Hobbs 1977]. This is because there are so many different ways for cloud to form in our atmosphere. Two of the most common types of clouds are convective
clouds and layer clouds. Convective clouds are formed by the local ascent of warm air
parcels in a conditionally unstable environment [Wallace and Hobbs 1977]. Convective clouds are also known as cumulus clouds and can be distinguished by their
predominantly upward growth. These clouds range in size from 0.1 to 10 km. Their ascension velocities can range anywhere from 1 m/s to 10 m/s, and they can last from minutes to hours [Wallace and Hobbs 1977]. An example of a cumulus cloud formation is shown in Figure 21.
Layer clouds, on the other hand, which are also known as stratus clouds, are
produced by upward motion of stable air and may occur anywhere from ground level up to the tropopause8. These clouds can extend from hundreds to thousands of square kilometers, are usually driven by vertical velocities ranging from 1 to 10 cm/s, and can last for tens of hours [Wallace and Hobbs 1977]. An example of a stratus cloud formation is shown in Figure 22.
Figure 22: Stratus cloud formation. Figure taken from [Wallen 1992].
In addition to cumulus and stratus clouds, clouds can also be classified as either
cirrus or nimbus. Cirrus clouds are those clouds which appear fibrous, and nimbus
refers to any cloud that precipitates. Most clouds can be classified by one of these types or their compounds, like cirrocumulus or cumulonimbus. Of course there are many more
documented cloud types than we will present here. However, for those interested more information can be found in the International Cloud Atlas, published by the World Meteorological Organization in 1956.
3.1 THERMODYNAMICS
Several thermodynamic principles are active participants in the formation of clouds in our atmosphere. One of the foremost and most relevant for our simulation model is the
ideal gas law, given in Equation 1.
nRT PV =
Equation 1. P is pressure, V is volume, n is number of moles, R is the universal gas constant9, and T
is temperature.
This equation tells us that, given constant pressure in a gas under ideal conditions, volume and temperature will be directly proportional. Or, given constant volume, the pressure and temperature will be directly proportional. Because of this relationship, a rising parcel of air will experience a decrease in temperature just as it experiences a decrease in pressure10. 9 Given as K mol J ⋅ 315 . 8 [Lide 1992].
3.2 PHYSICAL PROCESS
3.2.1 STAGE ONE: DIRTY MOIST AIR
The physical process of cloud formation as it occurs every day in our atmosphere is a quite complicated process. In order for the process to begin, the presence of “dirty moist air” is required. We must have air which contains tiny particles, called hygroscopic
nuclei, upon which the water vapor can condense11, along with having air saturated with water vapor (near 100% relative humidity12). Water vapor can actually condense at relative humidity levels lower than 100% if the air contains a high concentration of these particles (over 50%). For example, in a city with severe air pollution clouds may form with relative humidity as low as 76%[Wallen 1992]. Also, cloud formation can be artificially aided by increasing the local concentration of hygroscopic nuclei. In fact, this is the same concept used in artificially seeding real clouds — planes release fine particles into saturated air to aid in the condensation process [Wallen 1992]. An example of this is shown in Figure 23.
11 In order to condense, pure water must have a surface to adhere to.
12 Relative humidity is the amount of water within a differential volume of air over the total amount of water vapor that differential volume can hold (which is dependant upon pressure and temperature) [Wallen 1992].
Figure 23: Artificial cloud seeding over 45 minute period. Figure taken from [Wallace and Hobbs 1977].
3.2.2 STAGE TWO: ASCENSION
The second stage in atmospheric cloud formation is ascending air. Ascension is most often caused in our atmosphere by thermal currents. Thermal currents are the result of buoyancy within our atmosphere. The surface of the earth, warmed by the sun, radiates heat into the atmosphere, which causes the temperature of the air to increase. This causes the air to expand due to increased local pressure and then rise due to buoyancy13.
13 Buoyancy is a result of variation in density within a fluid. In our simulation, we do not account for local variations in density, only the resulting buoyant force.
A visual example of how temperature variations create thermal currents is shown in Figure 24.
Figure 24: Example of ascension caused by convection. Figure taken from [Wallen 1992].
Ascension of air can also be caused by orographic uplift which is when air is ‘pushed up’ by the terrain below it. For example, clouds often form around mountain ranges as the air blows up and over the surface of the mountain [Wallen 1992]. An example of this is shown in Figure 25.
Figure 25: A cloud formation caused by orographic uplift. Figure taken from [Wallace and Hobbs 1977].
3.2.3 ADIABATIC COOLING/SATURATION
As air rises, pushed by either buoyancy within the air or orographic uplift, the air expands as the pressure decreases. The decrease in pressure results in a proportional decrease in temperature. A graph of pressure (mBar) vs. altitude (km) is shown in Figure 26. Because cool air is able to hold less water vapor than warm air, the result is an increase in relative humidity as a moist air parcel rises. Eventually, the rising air will reach a point of saturation. The temperature at which this occurs is called the dew point.
Figure 26: Graph of pressure vs. altitude. Data taken from [Lide 1992].
3.2.4 CONDENSATION AND LATENT HEAT
Once the dew point has been reached and if hygroscopic nuclei are present,
condensation will occur. Since the process of condensation is an exothermic process, it causes the release of what is called the latent heat of condensation. This energy, which is proportional to 2.5×106 Jkg[Wallace and Hobbs 1977], increases the temperature of
the nearby air, thus increasing it's buoyancy and causing the air to rise even more. This is why so many clouds seem to grow upward, especially when the surrounding air is calm. Different atmospheric conditions contribute to the growth of different cloud types, which will be discussed later in section 5.3.
3.2.5 PRECIPITATION
The final stage of the process is precipitation. Precipitation is a gradual process involving the collision and coalescence of condensed cloud droplets. As droplets collide with each other within the cloud, many smaller droplets will combine to create a few larger droplets. As these droplets grow in mass, the gravitational force exerted upon them increases and thus the terminal velocity of these droplets increases. This increase in size and velocity also increases the probability of collision. Figure 27 below shows a comparison in size among cloud droplets and their larger counterparts, raindrops. CCN stands for Cloud Condensation Nuclei, and is equivalent to our concept of hygroscopic nuclei. In the figure, r is the radius in micrometers, n is the number of particles per liter of air, and v is the terminal velocity in cm/s.
Figure 27: Visual depiction of cloud droplet size. Figure taken from [Wallace and Hobbs 1977].
The statistical model of the precipitation process is called the continuous
of droplet size within a cloud based on a stepwise function of collision probability. The collision probability in the figure is 101 . One shortcoming of this model is that it fails to
consider increased probability associated with increasing size, although it can be argued that such a small change in size may not affect collision probability. As a point of fact, however, the complete collision probability solution is much more complicated,
involving not only accurate simulation of simple physical properties of cloud droplets such as mass, velocity, and acceleration, but also more complicated characteristics such as surface tension and body deformation. However, for the purposes of a visual
simulation, this model would present an adequate basis for a simulated model of the precipitation process.
4 FLUID MODEL
Our simulation of cloud formation is based upon an implementation of a course grid fluid solver presented recently by Stam [1999]. This solver uses the Navier-Stokes fluid equations to model fluid flow through a volumetric space. Here we present a brief description of the Navier-Stokes solution and why it is acceptable for our model. For details concerning our implementation, the reader should consult both Stam's paper [1999] and the paper by Fedkiw et al [2001].
4.1 FLUID MODEL BASICS
The Navier-Stokes solution is an approximation of incompressible flow.
Incompressible flow is that which contains no sharp changes in density, as would be caused by a supersonic shockwave. This model is preferred to those of compressible flow because compressible flow solutions require relatively small time steps to maintain accuracy. There is no disadvantage to using incompressible flow as long as the
maximum flow speed remains below Mach 1 (the speed of sound) [Brower 1999].
When building a model of fluid flow, we begin by assuming that mass is conserved. This means that the net change in mass must always be zero. This is expressed
change in density
(
∂∂ρt)
is balanced by the divergence (∇⋅ of the mass flux ) ( uρ ) at any point in space.t
u
=
−
∂∂⋅
∇
ρ
ρEquation 2
In Equation 2, ρ is the density and u is the vector field representing velocity within the fluid. If we assume that our density is constant (∂∂ρt
=0
), non-zero, and varies gradually compared to changes in velocity, Equation 2 simplifies to0
=
⋅
∇ u
Equation 3
This equation expresses conservation of mass within our system, and is required for the system to be properly constrained. We need four equations to solve for the four unknowns in a fluid system: velocity in x,y,z directions and pressure. Therefore
Equation 3 and the three Navier-Stokes equations are a complete solution for fluid flow. The Navier-Stokes equations can be written together as shown below in Equation 4 [White 1991].
ρ ρp
ν
f t u+
u⋅∇
u
=
−∇+
∇
u+
∂ ∂(
)
2 Equation 4The terms on the left of the equation are our inertial acceleration terms. The first term expresses our temporal acceleration while the second expresses spatial effects within the system. On the right of the equation we have a pressure term followed by a term which expresses the viscous force (shear stress). The final term, f, represents any external forces per unit volume applied to the system. This equation shows that motion within the system is dependant upon force due to pressure, force due to viscous effects, and force due to any external forces such as gravity (or buoyancy).
Because the inertial effects outweigh the contribution of viscous effects in air, we can ignore the viscous term. To see this we examine the Reynolds number for our system. The Reynolds number expresses the dimensionless ratio of inertial forces to viscous forces and is given below in Equation 5 [White 1991]:
νL u
R=
⋅In Equation 5 u is a characteristic velocity, L is characteristic length and ν is kinematic viscosity14. For air at approximately 20 °C, m s
2 5 10 4 . 1 × − = ν [Lide 1992].
Also, since the troposphere extends approximately 12 km from the surface of the earth [Wallen 1992], we can use 103m as our characteristic length. Therefore, in order for the
Reynolds number to be 1.0 (where the inertial effect equals that of the viscous effect), our flow speed would have to remain below 10−8ms. Since the fluid motion we will be
modeling will be much faster than this, we can assume that R>>1 and therefore viscous effects can be ignored. Although the precise value of the characteristic length (L) or kinematic viscosity )(ν may vary slightly in a particular simulation, this remains a safe assumption since the Reynolds number will remain much greater than 1. Looking at the Navier-Stokes equation again, we can see that, having removed the viscous term, our inertial acceleration in the system will be balanced by pressure and external forces within the system. This makes our fluid flow computation much easier since we do not have to compute the effects of viscosity.
The fluid solver we use for our simulation, presented by Stam [Stam 1999], is based on this Navier-Stokes system. At each simulation step, the Navier-Stokes equations — neglecting the viscous term — are solved for each voxel in the grid. One of the main aspects of this method is the use of backward-tracing particle motion within the system. This approach allows for unconditional stability even when using large time steps and is
14 The kinematic viscosity of a material is defined as the fluidic viscosity of the material over the density [White 1991].
shown in Figure 29. As shown in the figure below, at each timestep we trace the motion of a particle in the direction of it’s current velocity times −∆t. Instead of calculating where a particle is going, we are calculating what is coming toward this particle. This method introduces a small amount of inaccuracy, but makes the simulation much more stable and allows for larger timesteps.
Figure 29: Backward tracing time-step method proposed by Stam. Figure taken from [Stam 1999].
4.2 APPLICATION OF FLUID MODEL TO ATMOSPHERIC SIMULATION
In order to apply this fluid model to a simulation of the atmosphere, we must account for several discrepancies between the model and the actual atmosphere. As we have mentioned previously, the Navier-Stokes model of fluid motion assumes constant
pressure and density of air, which is not realistic in the atmosphere. Air in our atmosphere is stratified, meaning that its pressure and density decrease as we go up. Fortunately for us, these variances in pressure and density do not affect the motion of the fluid within the atmosphere, since the only direct affect of these changes would have on fluid motion would be realized as effects of viscosity (in our model viscous effects are ignored). What these changes in pressure and density do affect are the processes that occur at different altitudes within our simulation. For example, the pressure and density of air does have an effect on the amount of water vapor the air can hold, which directly affects the relative humidity computation in the simulation. Therefore, although we do not model local changes in pressure or density of the air, we do keep track of these values and how they change over large changes in altitude. By storing pressure and density values in lookup tables, we are able to use empirical data from the atmosphere to compute derived values such as relative humidity and how they change with varying altitude.