Organized Patterns of Movement in Space
Everywhere we look, we find an orderly arrangement of matter in the Universe, whether on the atomic level, or that of the galaxies; whether living organic life, or inanimate dead substances. With so many examples all around us in every arena of the Universe, it should seem preposterous that no orderly weather patterns have been found to date. You might ask, "What are the neat patterns in weather?" I asked that question, and began to explore some well known orderly patterns of moving bodies in other fields of science. First, I must say that I discovered the organization in weather systems by using empirical methods-if it works, use it, even if you don't understand why. After that discovery, I began to search for the theoretical foundation of my discoveries. The purpose of this chapter is to show some of the paths by which the theoretical explanations were tracked down.
The examples I use from other areas of science are only a small sampling of the myriad of similar types of examples that can be selected from the whole Universe. My discussion of some of these scientific features is not complete or exhaustive. The examples were chosen merely to illuminate certain well known and established laws of nature. These general laws of nature also serve as a statistical proof of the rules I use for weather patterns. Before we proceed with the examples, it is important to mention the priniciple of similitude.
The Principle of Similitude
The growth, decay, and shape of any structure is determined by the forces acting upon it. The impact of certain forces may be determined by the area of the structure. For example, the wind will push a large
sail on a sailboat with greater force than a small sail. The impact of other forces may be determined by the volume of the structure. An object (made of the same material) floating in the water will have greater buoyancy, the larger it is. In this example, the volume of the structure is the determining factor. We find that in similar figures, surface increases as the square, and the volume increases as the cube, of the linear dimensions. In the case of the sphere with a radius r, the area in the plane of a great circle is pi times r squared; the area of the surface of the sphere is 4pi times r squared;and its volume is 4/3pi times r cubed.. If the radius of the sphere is doubled, the area of the circle becomes 4 times as large; the surface of the sphere becomes 4 times as large; and the volume becomes 8 times as large. Since volume is closely related to mass or weight, we find that a fish that is doubled in length will weigh 8 times as much.
It was Galileo who first described what is known as the principle of similitude. He said that some of the forces acting in a system vary with the length, mass, and other factors; other forces vary with the square of these quantities; and still others as the 3rd power of the quantities involved. He also said that if we tried to build ships, palaces, or temples of huge size, that beams and bolts would refuse to hold together; and that a tree or an animal can not grow beyond a certain size and still keep the same shape. They would fall to apart of their own weight unless: (1) the relative proportions of their parts are changed (which would eventually make them clumsy or inefficient), or (2) they would have to be made of harder and stronger materials.
This principle is mentioned so that we can properly judge phenomena that vary or have a reason to vary when going from a small to a large size. We will now lightly touch on some astronomical and atomic laws that are well known; and then we will try to grasp those features that persist regardless of size to see if the same laws can be applied to the analysis of weather systems.
We start with Johannes Kepler (1571-1630) who discovered the three great laws of planetary motion: (1) The orbit of every planet is an ellipse with the Sun at one focus; this defines the shape of the orbit. (2) The straight line joining a planet and the Sun sweeps over equal areas during equal times. Therefore, the speed of a planet increases and decreases as it makes a complete circuit around the Sun. This law is
a consequence of the conservation of angular momentum. See Figure 5-1. Finally, (3) the squares of the period of revolution of the planets are proportional to the cubes of their respective mean distances from the Sun. The period is the time required for a planet to complete a round trip on its elliptical pattern. This is often called the Harmonic Law. _
It has been known for a long time that the distance between the sun and its planets seemed to follow a regular pattern. When these distances are compared, they more or less follow a law which was first discovered by Wolf in 1741, this was picked up by Titius in 1772, and finally popularized by Bode in 1778. This law of Wolf (called Bode's Law by some) goes as follows:
Establish the series of numbers beginning with 0, 3, 6, 12, 24, . . .; where each number is double the value of the number that proceeds it. You will have the following series:
Now the connection between these numbers is more or less the same as the one between the distance of the Sun and its planets. Taking the Earth's distance at 10, you can calculate the mean distance of every planet in proportion. Table III compares Wolf's series of numbers with the relative mean distance between the Sun and the planets. Wolf's
Is there also a systematic spacing between lows and highs on a weather map? Is there a similar formula?
Rings and Spokes of Saturn
Now another example in the solar system. The rings of Saturn are geometrically perfect, and they lie in the plane of Saturn's equator. They are the flattest known structure in relation to their thinness. The rings consist of many particles. It has been proven that the inner rings move faster than the outer ones; and at speeds made necessary by Newton's Law of Gravity and Kepler's Laws of Motion.
There are three main sets of rings, with the main controls on the shape and movement of the rings exercised by gravity and centrifugal force. My main interest was to see if there was any similarity in the type of mathematics involved in explaining the rings of Saturn and the movement of storms (in the middle latitudes of a hemisphere) in what seemed to be a ringlike drift around a pole. The Voyager I and II fly-by of Saturn did not alter the fact that the main forces are gravitational and centrifugal; but it did indicate that there are some small additional forces being brought into play to make the rings a little more complex. The discovery of the radial spokes traveling around in the rings of Saturn is intriguing, but geometrically speaking, there should be no surprise--radial spokes are a common occurrence in many types of physical phenomena, in spite of the differences in the kinds of forces involved. This is examined in Chapter 8: The Problem of Plateau.
One more thing before we leave the planets. It has been calculated that if a moon is closer to a planet than a certain distance, tidal forces
will break the moon up into small pieces. Oppositely, if fragments already exist at such a distance where tidal forces are destructive, they will not join together to form a single body. This limit is called the Roche limit after the astronomer E. Roche who discovered it in 1849. The Roche limit is generally equal to 2.44 times the planet's radius. The rings of Saturn do fall within the Roche limit.
The one common thread in all of these astronomical examples is that there are radial and/or circumferential spacings of some physical quantity as you move out along a radius from any common central object. It shows that (at least on an astronomical scale) that bodies or matter orbiting a central zone can not always take any radial position outward from the center, but are limited to certain fixed positions (or quantum positions). Quantum in the sense that any positions in between the fixed ones are forbidden.
Now let us shrink from the wide astronomical view to the tiny world of the atom.
Laws of the Atom
It is known that the electrons are arranged in systematic rings or shells around the nucleus of an atom. Niels Bohr (1885-1962) showed that all three of Kepler's laws for the planets also hold true in the case of the electron circling around the nucleus of the hydrogen atom. In addition, he showed that there was a quantum law, that the electrons can revolve only in certain fixed orbits and in no others. This is a harmonic law similar to the law that limits Saturn's rings to exact size and shape.
Bohr's model of the atom consists of 7 shells or rings. These are quantum rings inasmuch as no electrons can occur in between any two consecutive rings. Table IV shows the number of electrons possible for each ring, which is equal to the ring number squared times 2.
The diameter of these rings also follows a rule of squares and is measured in Angstrom units, which is the unit of length (equal to one hundred millionth of a centimeter) used to measure light waves. Table V gives the diameter, in Angstrom units, of the 7 rings of the hydrogen atom.
The hydrogen atom has only one electron. The ring that the electron occupies depends upon the amount of energy it receives. It will jump from one ring to another when it has the required changes in the
amount of energy it receives. The electron never circles in between the rings, but it always jumps the whole distance or not at all. This is in accord with the laws of quantum mechanics for the atom.
With this information, Bohr was able to calculate the speed of the electrons. In the first four rings of the hydrogen atom, he found that the electron moves at 2160 km/sec., 1080 km/sec., 720 km/sec., and 540 km/sec., respectively. These speeds are related in the exact proportion of 12:6:4:3.
We come to the conclusion therefore, that the orbit of the electron in a hydrogen atom is equal exactly to one complete electron wave joining to itself around the ring.
To understand this a little better we will look at the vibrations of a ring made of wire. You will see that the number of wavelengths always
fit an integral or whole number of times into the ring's circumference. Each wave connects perfectly with the next one. The wavenumber is the number of whole wavelengths that fit into the circumference of a circle. In Figure 5-2, we show only wavenumbers of 1, 2, 3, 4, 5, and 6. If there were no resistive or dissipative effects, these waves would persist indefinitely.
You will find that you can not fit a fractional number of waves into the ring and still have each wave join the next one smoothly. Fractional wavelengths would set up destructive interference as the waves go around the loop, and the vibrations would die out quickly.
An electron can circle an atomic nucleus only if its orbit is a whole number of electron wavelengths in the circumference. As the electron jumps to a higher ring, we find that the size of the ring increases. The increasing circumference of the higher rings makes it possible for an increasing number of electron wavelengths to fit in. The circumference of the elliptical orbits of the hydrogen atom can likewise only be broken up into whole wavelengths.
The Bohr model is a crude approximation, since the electrons are actually considered to be cloudlike waves over a three dimensional volume of the atom; what we have been considering is the two dimensional view.
We see that there are many similar structures, shapes, and rules that apply to the confined space of the atom and to the huge playground of the solar system and the Universe beyond. Rotating high and low pressure centers are somewhere in between these two extremes. The concept of wavenumber has been recognized for a long time in the meteorological literature. There is usually some reference to wavenumber every month in professional journals-but these references are always referring to the average wavenumber on a weather map for a week, a month, or a year; never a precise calculation for a specific weather map for a given moment in time. In addition, the concept of wavenumber has been limited to the measurement of the
number of wave lengths of highs and lows along the circumference of a given latitude circle (or in a range between certain arbitrarily chosen latitude circles). You shall see later that this is an artificial limitation when I introduce the principle of staggered wavenumber around a point, in addition to the regular wavenumber as defined around a circle. In Figure 5-3 we see a liverwort showing one ring of 8 short arms or tentacles and a second ring of 8 longer tentacles. We can see that there are two rings, each with a wavenumber of 8. If we join the ends of the tentacles in any one ring with straight lines, we will get 8 equal wavelengths. We can also join the ends of the tentacles alternately to the inner ring and then to the outer ring as shown by the dashed lines to give a staggered wavenumber of 16. In actual weather situations we may find parts of the inner ring missing, or parts of the outer ring missing, as we shall see later on.