The purpose of this Webcast is to help you understand the basics of dispersion meteorology so that you can better respond to an event in which hazardous materials are released into the atmosphere. The National Weather Service (NWS) can have certain responsibilities during an emergency that are described here:

1. You may be asked to provide a site-specific wind observation or estimate.
2. You may be asked for atmospheric transport and dispersion information.

In these kinds of events, you will probably need to call Air Resources Laboratory (ARL), give them information for a model run, help emergency managers interpret the model results, and use your own expertise to evaluate how the local meteorology might affect the model results and whether these conditions are likely to change.

- Call the Senior Duty Meteorologist at NCEP (SDM, 301-763-8298, fax 301-763-8592, e-mail sdm@noaa.gov)
- Provide information such as location, size, and/or height of release
- Get model results from secure Website

One of our goals is to help you understand what goes into a dispersion model, rather than just thinking of it as a black box that spits out answers. I'll talk about how a dispersion model calculates concentration (and exposure) by focusing on two main dispersion concepts: plume rise and turbulence, and I'll describe some of the different types of dispersion models. I'll also mention some of the limitations of models and complications that can affect model results.

We use dispersion models to evaluate whether or not the public has been, or is going to be, exposed to hazardous materials. That exposure is a function of three things. The first is dispersion, or how well-mixed the plume is. A plume disperses (or dilutes) in both the horizontal and vertical, and we'll talk a little later about some of the factors that determine whether a plume is relatively dense or more dispersed.

Exposure is also a function of how close a person is to the centerline of the plume, and this is largely related to plume rise. The next section of the Webcast will discuss what causes and affects plume rise.

Finally, exposure is also a function of time (i.e., how long a person is in the plume), which depends on how constant the wind is, how long material continues to be released into the atmosphere, and where it is transported.

Now let's talk about some concepts related to dispersion, also called diffusion. We use these terms interchangeably—both refer to atmospheric, turbulent processes.

Dispersion is assumed to be random (also described as a Monte Carlo or drunkard's walk process). This kind of process is similar to molecular diffusion. But the atmosphere is not really randomly diffusive; it tends to diffuse in different directions at different rates and is, of course, strongly influenced by wind direction and wind speed.

Over time, the dispersion process will create an essentially normal distribution of material in both the horizontal and vertical. One illustration of the importance of time that I've used in classes is to have one student in the first row, two in the next row, three in the next, and so on. The first student flips a coin and depending on whether it's heads or tails, passes it back to either the right or left. You can visualize what happens with just one coin—when it gets to the back row, it could be held by any of the 7 or 8 students in that row. But if you let the experiment go for, say, 20 minutes, at the end you have most of the coins held by students in the middle and fewer coins held by the students at the end—a normal distribution.

One other thing I want to point out is that diffusion processes always act to dilute; a plume never re-concentrates in this process.

Dispersion is a dilution process. Basically, the more air you mix with a plume, the more dilution you have and the more the plume disperses. This is similar to what you would see if you poured dye into a river and watched how the color changed as the water moved downstream.

Dispersion is primarily governed by turbulence, which mixes ambient air with the plume. The three kinds of turbulence that act to disperse a plume are mechanical turbulence, turbulence caused by shear, and turbulence caused by buoyancy.

Mechanical turbulence is caused by air flowing over rough features, such as buildings, trees, and hills. Turbulence from shear can result from differences in speeds (such as between the plume and the ambient wind) or direction. Buoyancy turbulence can be caused by something as dramatic as an explosion or as simple as bubbles of air rising during the diurnal heating of the surface. Particularly in the latter case, buoyancy is governed by the stability of the atmosphere.

Stability is the degree to which the atmosphere will support, tolerate, or suppress turbulent motions. The basic stability criteria are familiar to you; an unstable atmosphere is one in which the lapse rate is superadiabatic, in a neutral atmosphere it is roughly adiabatic, and in a stable atmosphere is less than adiabatic. Stability has a stronger influence in the vertical than in the horizontal, and you'll see this later when I show a diagram of how plumes spread under different stabilities.

In dispersion modeling, we use more direct measures of stability than the stability criteria. There are several ways to measure turbulent dispersion. One is sigma theta, which is the standard deviation of the horizontal wind direction fluctuation, and it's commonly measured with the kind of wind system shown to the right. The other is sigma w, the standard deviation of the vertical wind speed fluctuation. The little propeller shown in the figure at left is typically used to measure these changes in the vertical wind.

Another way to estimate turbulent diffusion is to calculate friction velocity. Although the equation looks complicated, it's really just a function of two things: the wind shear, du/dz, and the mixing length, l-sub-z, which is how far a parcel of air will move if it's displaced up or down, and which depends on the stability of the atmosphere.

This equation can also be used to predict the wind profile in the boundary layer, and this results in what's commonly called the log wind profile.

You may have run into the five classic plume types in classes or text books, but I want to remind you of them and how they relate to stability. The first is the fanning plume, and on the left you can see that the dashed white line is the adiabatic lapse rate and the dark line shows that the actual lapse rate is very stable. A fanning plume tends to be very narrow in the vertical. Over a short period of time, it's also narrow in the horizontal, but as the wind direction fluctuates, it tends to spread out widely in the horizontal while staying very confined in the vertical.

The second is the lofting plume, where you have a stable layer underneath a neutral or unstable layer so the plume is lofted upward. It can't disperse downwards because of the inversion and stable layer, so you get this look with a flat bottom and a rising plume on the top.

The third plume type is called looping. Here you can see there is a super-adiabatic lapse rate from the ground up to plume height, and the plume goes rapidly up and down as it goes through thermals. The drawing is a snapshot, but if you looked at this over time, you would see the plume spread very widely over the vertical.

The coning plume occurs when there is roughly a neutral lapse rate from the surface well past plume height. Here the plumes grows gradually both upwards and downwards, resulting is this cone shape.

The last plume type is the fumigating plume. It's a special case of the fanning plume that goes through a transition. Imagine that a fanning plume, which is very stable and very confined, extends a significant distance out over the countryside. And then as mid-morning comes, the stable layer begins to erode and as it gets to the plume level, it mixes the plume down to the ground in a fairly concentrated amount. The important thing is to realize that this process can extend a concentrated plume a significant distance from the source and then rapidly mix it to the ground, which can be kind of a surprise.

This section is about plume rise and wind. You might wonder why plume rise is so important, and I think the diagram on this page illustrates this very well. This diagram represents a famous photograph of two plumes from a power plant in Salem Massachusetts. The shorter stack is 250 feet and the taller stack is 500 feet, and you can see that in such a short distance, the winds at the two heights are going in nearly opposite directions. You can see that, if you didn’t get the plume rise right, you would have no idea where the plume had actually gone.

As I was preparing for this Webcast, I looked up a number of plume rise equations, and in just an hour I found about 10 different ones. So I decided to make this easy. All of the equations have the same basic relationships. Plume rise is directly proportional to the ejection velocity (how fast the plume is coming out) and plume buoyancy (a function of how hot the plume is). And it is inversely proportional to the wind speed (the higher the wind speed, the lower the plume rise) and to stability (the more stable it is, the lower the plume will be).

 

There are four important things to understand: The ejection velocity forces a plume into the atmosphere, but its effects are quickly diluted by actively entraining ambient air. As long as the plume temperature is significantly greater than the ambient temperature, the plume will continue to rise. Light winds allow the plume to stay intact as it rises, while strong winds will bend the plume over and quickly mix it with the surrounding ambient air so that it becomes neutrally buoyant more quickly than with light winds.

The classic plume examples that we've illustrated throughout this Webcast have been from stack sources such as a power plant or some kind of manufacturing facility, but you may be called upon to deal with some special cases.

Spills from pressurized containers, such as a chlorine tanker, may have an ejection velocity, but it may not be pointing up. So you need to know which way it's pointing.

Fires don't have an ejection velocity, but can be very buoyant. For example, you've probably seen pictures of forest fire plumes rising as high as 30,000 feet.

The case of dense gases is a very unique situation. A dense gas is one that is cold or more dense than air. You need to think of these plumes as behaving as a liquid. If there is any significant slope to the terrain, the dense gas plume will move downhill rather than with the prevailing wind direction.

Let's talk about a specific situation where plume rise is important. Imagine an oil refinery fire that starts in late afternoon when the atmosphere is neutral and the winds are light. The hot plume rises and passes over the nearby town.

But what happens if your forecast is for much stronger winds by early evening? While a stronger wind is more dispersive, the plume would lay down much closer to the ground because the plume rise would be lower, and the concentration in the town could rise dramatically. You would probably want to warn your local officials about this possibility.

During a typical evening, a surface inversion develops, and the town might be protected by a surface stable layer. This means that the plume cannot disperse downwards as much as upwards, and often concentrations near the surface are less.

Plume rise is one part of a dispersion model calculation, and now I want to talk more about dispersion models in general.

We're all aware of different approaches to modeling physical processes. We have the Eta model, the NGM, the AVN, for example. There are many kinds of dispersion models, as well. We'll focus primarily in this talk on three: the Gaussian Model, which assumes a normal distribution and is the most common type of model used; the Gradient Transport Model, which uses K theory (and I'll explain that a little later); and the Puff Model, which is used for non-linear emission sources. I'll just mention three other types of models: statistical models, which are based on the Monte Carlo path approach; similarity models, which are commonly used in fluid mechanics and wind tunnels; and box models, which are often used for urban diffusion problems.

The Gaussian dispersion model is a simple way to approximate the dispersion process, Here's the classic normal curve, and I want to point out that this normal distribution occurs only after some time, as in the coin toss example I mentioned earlier.

If you go 2.15 sigma or 2.15 standard deviations out from the curve in both directions for a total of 4.3 sigma, you comprise about 99% of the area under the curve. A pollutant plume has a normal distribution in both the horizontal and vertical, and if you go out 2.15 sigma in both directions on both axes, again for a total of 4.3 sigma, you will have encompassed 99% of the plume's mass.

When you want to predict a plume's dispersion, you need to have some way to predict the dispersive capacity of the atmosphere in both the horizontal and vertical. I mentioned earlier that one way is to measure sigma theta and sigma w, but if those measurements aren't available, a very simple way is to use what are called the Pasquill turbulence types, shown in the table. The various stability categories are a function of time of day, sky cover, and wind speed. For example, during the day, when you have clear skies and very low wind speeds, the stability category is A, which is extremely unstable. Under these conditions you would have strong vertical motions that would disperse the plume significantly. At night with clear skies or light cloud cover and winds 2-3 meters per second, the category would be F, or very stable. This plume would be more confined in the vertical, but over time would spread out in the horizontal.

Once you have the stability category, you can use the equations on the right to predict sigma y, the horizontal dispersion, and sigma z, vertical dispersion. Remember that sigma y and sigma z represents one standard deviation out from the center of the normal distribution.

If you plot these equations, you get the curves you see here. The horizontal curves are roughly logarithmic, while the vertical dispersion curves are quasi-logarithmic and diverge depending on stability.

Basically when the atmosphere is stable, the normal curve is steep, but flat when it's unstable. Also, stability strongly suppresses vertical motions, while horizontal motions tend to spread the plume very widely even when it's stable.

This diagram illustrates the plume geometry for the y- and z-axes. A plume is always traveling downwind in the x-axis, and you have a normal distribution in both the y and z directions.

I thought it would be useful to show you the basic equation for Gaussian dispersion. Q is the source strength, usually in grams per second; sigma y is the standard deviation in the horizontal (and we just saw how to calculate that from the equations), and sigma z is the standard deviation in the vertical (also coming from those equations). U is the wind speed and 2 is in the denominator. This equation predicts the centerline concentration of a plume. In the situation of a tall smoke stack, the centerline and maximum concentration is well above the ground. For a surface release with no plume rise, the centerline and maximum concentration is at the surface.

This is a diagram of a typical plume configuration and you're looking down on the source. You can see that the dashed lines in a circle represent where the stack is. We place a virtual source a little upwind to simulate the fact that the plume is diluted as it comes up from the stack then goes downwind. You can see that 2.15 sigma is the dimension in each direction that when totalled along both axes represents 99% of the plume mass.

This is the same diagram, only looking from the side. The vertical dimension is sigma z, and again the virtual source is a little upwind to simulate the effects of plume rise.

I like to see real numbers in an equation, so here's an example. Q is the source strength of 100 g/s of sulphur dioxide; u, the wind speed, is 10 m/s; and for a receptor located 1000 m downwind (and on the centerline of the plume), sigma y from the curves shown earlier would be 76 m and sigma z would be 38 m when the stability is D, or neutral.

When you do the calculation, you come out with 0.00055 g/m³ or 0.21 ppm. I gave this presentation to a class and they asked "Is that good or bad." That wasn't really the purpose of my example, but I can tell you that 0.21 ppm is about 40% of the national ambient air quality standard for sulphur dioxide.

This is the entire Gaussian plume dispersion equation—don't worry, you won't be tested on it. This expanded form gives you the concentration at any point, while the previous equation was for just locations along the centerline of the plume.

Concentration falls off rapidly in either the y or z direction as it is a logarithmic, or quasi-logarithmic relationship. And one important consideration is that as H, height, approaches zero, concentration at the surface will double. Physically, this is because there can't be dispersion into the ground, and so mathematically we put two plumes, one at height z and one at height negative z, and when they merge together at H = 0, they cancel the 2 in the denominator and the concentration doubles.

So what do you need to know about Gaussian models? I think it's important to realize that 1) They represent an idealized solution. 2) That more stable means more concentrated at plume height, while less stable means less concentrated at plume height. 3) Concentration is inversely proportional to wind speed.

Those are the basics, but when are some of these assumptions not true? Well, the Gaussian model isn't really appropriate for long-range transport, (e.g., a volcanic ash problem). As a plume gets bigger at longer distances, the dispersive turbulent eddies are much smaller than the plume and no longer act to disperse the plume. And the other thing is, the concentration someone is exposed to isn't directly a function of stability, but rather is highly dependent on where they are in relation to the plume in both the horizontal and the vertical.

Another type of model is the gradient transport model, sometimes known as Fickian diffusion. In this model, the dispersion is proportional to the concentration gradient or the eddy diffusivity. Let me give you an example. Imagine a metal bar and at one end you have a heat source like a Bunsen burner. If you heat the bar a little bit, heat will migrate down that bar very slowly. But if you turn up the heat and heat it to glowing orange, heat will migrate very quickly. That's because the temperature gradient has been increased dramatically.

The analogy to eddy diffusivity is the heat capacity of that bar. If you use a metal bar, it will transfer that gradient of heat very quickly. If you use something that doesn't move heat quickly, like asbestos, then the eddy diffusivity would effectively be low. Asbestos would allow that gradient of heat to exist for a long time, while the metal bar does not.

So looking at the equation, the change of concentration with time is proportional to the concentration gradient and to eddy diffusivity. How do we calculate eddy diffusivity? Well, it's a function of turbulence, and the growth of the plume with time (as expressed by sigma y and sigma z) is proportional to the eddy diffusivity multiplied by time.

So what do you need to remember about the gradient transport model? Dispersion is proportional to the gradient in concentration. Concentration is proportional to stability and inversely proportional to wind shear and surface roughness. Over time the gradient transport model can give you roughly the same solution as the Gaussian model, but it's more useful for numerical modeling and more sophisticated applications. Dispersion can be calculated rather than parameterized, which is nice. And it's often used for regional models and long-range transport.

A puff model is used with an essentially instantaneous source such as a sudden release or explosion. Mathematically, this involves what we would call relative diffusion, while the Gaussian model applies to a continuous source. Relative diffusion occurs in a Lagrangian framework where the diffusion occurs in relationship to other parts of the puff, but is independent of the how the entire fluid is moving. Sort of like how boats might scatter from each other as they all drift downstream in a river. Puff models are also used to represent cases where the wind field is very variable and/or the emission source varies with time.

We've talked extensively about the Gaussian dispersion model and a little about the gradient transport model. There's a third model called the puff model. Here you can see the equation for the puff model, and basically it's the equation for a sphere that grows by radius sigma. It's relatively independent of wind speed, and it's a function of two parameters: the source strength and the standard deviation, which is a function of turbulence. An interesting subtlety here is that the puff diffusion is a function of eddy size. Eddies relatively larger than the puff are not very dispersive. If you have a very small puff, for example 10 m in diameter, and a relatively large eddy, say 30 m in diameter, the eddy picks up the puff and moves it to a new location with no dispersion during that process. Eddies smaller than the puff disperse it by grabbing parts and entraining ambient air. So a puff disperses more rapidly after it has grown larger than the eddies carrying it.

Finally, I'd like to talk about some complications in dispersion modeling, and some of these have to do with how your local mesoscale meteorology will affect the performance of a dispersion model. The models that we've discussed will not have the local knowledge that you do, so your expertise will be very important in interpreting and fine-tuning the model results. The models typically assume a single wind direction and do not resolve local features like drainage flows.

Two good examples of special situations are complex terrain and urban areas. We know that these can cause very unpredictable flow, they can increase turbulence, and they can bring people into contact with concentrated plumes. For example, if you're on top of a hill as a plume is going by, you might be more exposed than if you are in the valley. These features are generally not resolved in dispersion models.

I suspect that many of you have encountered the Froude number as you think about cold air damming or barrier jets. We use the Froude number in dispersion calculations to determine if a plume is going to go over a hill or around it. If the Froude number is less than one, we know that there will be some height where the plume will not go over the hill, but will go around it. Basically the Froude number relationship says that in light wind and stable conditions, the plume may be forced around a terrain feature instead of over the feature.

In situations where the atmosphere is neutral or unstable, we know the plume is likely to go over the hill rather than around. In neutral conditions it will go very high over the hill, and in unstable conditions it's likely to be much closer to the hill.

This graphic illustrates another complication where a source is immediately downwind of some terrain. You can see that, even though the upper level flow might be strongly from one direction, the flow in the lee of the hill might be from the exact opposite wind direction. The important point to take from this discussion is that you may know of terrain features, sea or lake breeze features and the like in your forecast area that will affect the dispersion and transport of the plume, and which are unlikely to be included in the prediction of a dispersion model. This is particularly important information that you will bring to the prediction process.

Dispersion in urban areas can be different than what's predicted by the dispersion model for a variety of reasons. Turbulence is often enhanced by the urban surface roughness of buildings, and these buildings can act similar to the terrain in the previous example by mixing a plume back to the surface through lee effects. Wind directions are often more variable. The urban heat island at night can easily change stable conditions to neutral or even unstable conditions.

I'd like to mention one other special situation, and that's the case of particulate plumes. These can result from fires, explosions, volcanic eruptions, or high winds kicking up dust. We know that particles will settle, and they will eventually stratify the plume by particulate size, with smaller particles higher up and heavier particles near the bottom. This table gives you some examples of the settling velocities. For example, a 50 micron particle could settle at 60 cm/s or0.6 m/s. A typical particulate plume 100 m above the ground with a wind blowing 4 m/s will have half of all the 20-micron particles settled out in about 8 km. Remember that particulate plumes are also easily scavenged by precipitation.

Things you need to think about when an incident happens:

1. How high is the plume (is it buoyant, is it a dense gas, is it being ejected), and what is the stability?
2. How does the current stability affect the plume's behavior?
3. How do you expect the stability to change over the course of the event?
4. Will plume material be able to mix to the ground where people will be exposed?
5. How are the wind speed and direction affecting the plume? How will this change during the event?
6. How will terrain and/or other surface features affect plume behavior?