Problems With Weather Prognostication: Boundaries Are A Bane; Initial Conditions Irk; Vertical Velocity Vexes

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Home / Problems With Weather Prognostication: Boundaries Are A Bane; Initial Conditions Irk; Vertical Velocity Vexes
It is impossible to precisely model the atmosphere when it has a lower boundary like this. Photo credit: ImBooToo

It is impossible to precisely model the atmosphere when it has a lower boundary like this. Photo credit: ImBooToo

St Swithun’s day if thou dost rain
For forty days it will remain
St Swithun’s day if thou be fair
For forty days ‘twill rain nae mare

Friends sometimes ask:

What’s the matter with you guys? Why can’t you tell me what the weather will be tomorrow? Next week? Saint Swithun’s Day? How hard can it be?

Stephen Curry shoots the ball from 25 feet away from the basket and gets it in half the time; the other half are pretty close. I forecast rain and it’s sunny — not close. And that’s for two days from now.

Next January? Not a chance to do any better than the climatological average. So what gives? Why is it so hard to forecast the weather?

Billiards And The Atmosphere: The Numbers Game

Let’s look at a good pool player. He deals with 16 balls and a table with known dimensions. The table has boundaries, and the player knows the properties of the boundaries. In particular, he is very confident he knows how a ball will bounce off the side boundary. He also knows the effect of friction between balls and the table, and the characteristics of interactions between balls.

So he is able to strike the cue ball, have it bounce off cushions and other balls, with the result that everything ends up where he wants it to.

Isn’t weather forecasting the same thing? Molecules bouncing off each other and off boundaries (mainly the surface of the earth or things attached to it like skyscrapers)?

Yes, but there are many more than 16 molecules to keep track of. How many? Luckily I have an unused envelope on which to calculate. The calculation will be approximate, because the numbers get very large and you’ll easily get the point.

A cubic foot of air contains about one and one-quarter liters.

One liter has, at room temperature and average surface pressure, one mole of gas, and one mole contains Avogadro’s number of molecules.

Avogadro’s number is about six times 10^23 — that’s six followed by twenty-three zeroes.

So the number of molecules in a square foot of air is, more or less, 10^24. If air molecules were shoes, this number would make Imelda Marcos’s closet look empty.

Maybe We Don’t Have To Deal With Individual Molecules: A Simplification that Reduces The Numbers

weather forecasting

Why is it so hard to forecast the weather? Possibly due to the huge amount of air involved… Copyright image by Decoded Science, all rights reserved.

Meteorologists are given to talking about ‘parcels’ of air. These can be various sizes and there’s no such thing as a parcel that acts like a unit, but they give scientists at least a chance of modeling the atmosphere.

A cubic foot is a convenient size for a parcel of air, so let’s see what the numbers look like if we use the cubic foot as the basic unit.

The area of the surface of the earth is approximately 200 million square miles and there are approximately 28 million square feet in a square mile.

The atmosphere is not very deep, vertically, and most weather takes place within ten miles of the surface. So the height of the atmosphere is about 50,000 feet.

If we multiply all the numbers together, there are approximately 10^20 cubic feet of air in the atmosphere. Well, it was a nice try, but there are still too many moving parts. In fact, the basic parcel of air has to be much bigger than a cubic foot, and the approximation to the real atmosphere will be quite crude.

We can measure the temperature and velocity at various points and interpolate on a grid. As measuring instruments become more precise and the number of observing stations increases, we have a chance to continually refine our knowledge of the state of the atmosphere at any given time.

When The Forecast Begins: Initial Conditions

If we want to make a forecast, we have to start somewhere, or rather sometime. Unlike the billiard player, who knows exactly where each ball is before he strikes the cue ball, we don’t know that our observations are taken at the same exact moment.

Worse yet, we still have vast numbers of air parcels whose values we can only infer from the observations we have. Initial conditions will always be a problem in any model.

Tiny disparities can change the weather tremendously - this makes forecasting the weather much more difficult than predicting the path of a billiard ball. Copyright image by Decoded Science, all rights reserved.

Tiny disparities can change the weather tremendously – this makes forecasting the weather much more difficult than predicting the path of a billiard ball. Copyright image by Decoded Science, all rights reserved.

As If Things Weren’t Hard Enough: The Butterfly Effect

It is easy to assume that if the initial conditions contain an error, a forecast will have an error which grows at a slow, steady rate. But that is not the case.

Errors in initial conditions explode in forecasts, and the result is the butterfly effect: An error as tiny as that created by a butterfly’s flapping its wings will cause forecasts to diverge from actual conditions at a rate that makes them completely useless within less than a week.

Boundary Conditions: Another Intractable Bugaboo

It is an unfortunate fact (I’m looking at this from a forecaster’s point of view) that we live at the surface of the earth. The surface is irregular with regard to composition and shape, and this introduces enormous problems to forecasting.

Forecasts for thousands of feet above the ground are much more accurate than those for the surface. But before you say ‘Phooey,’ realize that certain jet stream patterns correlate well with surface weather, and the upper air forecasts are a huge help in modern medium-range forecasting for ground-dwellers.

Since every hill and valley, nook and cranny of the earth’s surface is impossible to incorporate into a forecast, about the best we can do is to assume roundness and uniformity, or adjust the observations to reflect some semblance of it. This leaves much to be desired, since butterflies could be hiding anywhere.

Precipitation: Hardest Of All To Predict

We are basically interested in two aspects of the atmosphere: temperature and precipitation. The former is much easier to forecast to the satisfaction of the casual citizen, because the values fall within a narrow range — departure of more than ten degrees from the average is quite rare; twenty degrees is normally a record.

But precipitation comes in globs. Most of the time there’s none; sometimes there’s a little; and all too often there’s a lot. Furthermore a deviation of several degrees from the forecast is not normally more than a minor nuisance. But a downpour when none is expected can snarl traffic and douse pedestrians.

Why Precipitation Is So Hard To Forecast

This air parcel's trajectory in 2005 is typical. It traveled from Colorado to Louisiana and rose about a foot. Vertical motion is key to precipitation and is hard to forecast because it is small relative to horizontal motion.

This air parcel’s trajectory in 2005 is typical. It traveled from Colorado to Louisiana and rose about a foot. Vertical motion is key to precipitation and is hard to forecast because it is small relative to horizontal motion.

There are two factors in precipitation: Moisture and vertical motion. If there’s no moisture, there can’t be any condensation. That part’s easy. We just measure the amount of moisture in the air — the most convenient measure is the dew point, the temperature at which the air would be saturated. If the temperature falls below the dew point, the air cannot hold all the moisture and some of it condenses.

The ideal gas law relates pressure to temperature and volume. As pressure falls, temperature falls and volume increases.

In the atmosphere, pressure decreases with height (there’s less air above to press down), so if a parcel of air is lifted, its temperature also decreases as it rises. If the temperature drops to the dew point, condensation begins; rain or snow follows.

Here’s the problem: The vertical motion in the atmosphere is normally a tiny fraction of the horizontal motion; it is hard to detect, and even harder to predict.

Weather Forecasting Is Hard

A pool table is stationary; it has 16 balls and a surface and sides with known properties.

The atmosphere has a number with many zeroes worth of molecules or parcels. The earth is spinning (we never even got to that complication), the sun’s heating is different in different places (or that). The atmosphere is a continuous medium with no beginning or end, yet we have to provide initial conditions; the surface of the earth is not uniform, but we have to impose boundary conditions. Water vapor condenses, evaporates, sublimes.

There are butterflies everywhere. All things considered, it’s a wonder weather forecasts are any good at all.

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