temperature shown in figure 1, then it tends to remain in its superior position, and is in stable equilibrium; but if above the given elevation the remaining atmosphere is colder than the temperature shown by the curve of equilibrium, then the whole atmosphere is in unstable equilibrium and would be likely to be overturned.

The final condition to be considered and the most complex is an atmosphere containing water vapor. There is, of course, an endless variety of cases depending upon the initial temperature condition and degree of saturation. One case only will be considered in detail; perhaps it is the simplest, but it may serve in a general way as an illustration of all related cases. Assume as an initial condition a large mass of saturated air at sea level at a temperature of 80 degrees, a rather extreme condition, but one that may, perhaps, be reached in summer. Assume that some impulse starts this mass into upward motion into layers of air whose temperature and density vary according to the average observed conditions as shown in figure 1. As the saturated atmosphere rises into regions of reduced pressure, it will expand, and thus perform work. Its temperature will be reduced by the abstraction of sufficient heat to perform the external work. Since the air is originally saturated, lowering the temperature implies the condensation of some of the water vapor into the liquid state. Now, to lower the temperature of one cubic foot of saturated air one degree requires the removal of much more heat than would be necessary if the air were moisture-free; or, what is the same thing, to supply a certain amount of heat energy, the cubic foot of saturated air will have its temperature reduced only a fraction of the amount by which a cubic foot of moisture-free air would be reduced. As a consequence, the temperature of the rising moist air will be decidedly higher than that of the surrounding layers of stationary air.

The condensed moisture in this case will first appear as fog. If the particles are small enough they will fall at an exceedingly slow rate. As their size increases their vertical velocity will likewise increase. For the sake of simplicity let it be assumed that the condensed moisture is removed by falling as soon as it forms. On this assumption the temperature can be calculated for each pressure and thus for each altitude. This temperature is shown on figure 1, Curve D. This is then an irreversible process, due to the removal of the condensed moisture, and if through some agency the saturated air should be caused to fall, it would return to its original elevation along a curve parallel to Curve C. It is supposed that fog or cloud particles are usually between 0.001 and 0.0002 inch in diameter, and that a particle of the larger size falls through the air at a rate of 2 inches per second or 1 mile in 9 hours.

The distance between Curves B and D, showing the difference in temperature at any elevation, is in a way a measure of the difference in weight or density of air under the two conditions and so indicates the buoyant force. In hot summer weather warm air from the ground does rise every clear afternoon under such forces producing the cumulus clouds so constantly visible. Such vertical currents are called convection currents. When the supply of warm moist air is sufficient, a thundershower often results. By means of measurements on the clouds it has been determined that these ascending currents often rise to an altitude of several miles.

For purposes of comparison, Curve E has been added on figure 1, similar to Curve D except starting with a temperature of 70 degrees at sea level.

One point remains for discussion. In drawing Curve D, it was assumed that all condensed moisture was removed from the air as soon as formed. Actually, condensed moisture in the form of cloud or fog remains largely with the air in which it is produced. This cloud or fog increases the weight of the air containing it and so reduces the buoyant force. To indicate something as to the amount of this effect Curve F has been added showing by its position between Curves B and D the relative decrease in the buoyancy if none of the condensed moisture had been removed from Curve D. Curve G has been similarly drawn with respect to Curve E.

The data in table 1 and the curves in figure 1 show the nature and importance of the disturbing forces introduced into the atmosphere by the presence of water vapor. When a mass of air at the surface becomes well warmed and well saturated with moisture, a state of unstable equilibrium is reached whose disturbing effects upon the winds and weather often results in thundershowers and other forms of precipitation. The convection currents which result from such a state of unstable equilibrium seem to play an important part in all storms, but other factors also have a large influence, and no one yet has been able satisfactorily to show just what weight in producing precipitation must be ascribed to the effect of convection.

MOTION UPON THE EARTH'S SURFACE

In order to comprehend the most fundamental relations existing between the horizontal motions of the atmosphere upon the earth's surface and the forces which control these motions, it is necessary to have in mind some of the underlying laws of mechanics. The more elementary of these laws will be stated and illustrated briefly using graphical methods so far as possible to illustrate the relations existing between the differential quantities. For a complete mathematical

treatment by analytical methods the reader is referred to the treatises enumerated at the end of this chapter.

CENTRIPETAL FORCE AND RADIUS OF CURVATURE

In figure 2 let P represent at the beginning of a certain instant of time the position of a body moving in the direction of PA with a velocity V. If no force acts upon the moving body it will move in a straight line with unchanging velocity. This law of motion is commonly said to result from the · body's possessing the property of inertia. In a brief time, dt, the body moves a distance equal to V dt.

Vdt.dx

oldt-dy

Let

PA = V dt

(1)

2.

RA

FIG.
DIUS OF CURVA-
TURE.

Now suppose that during this same time, dt, the body is acted upon by a constant force, F, acting always at right angles to the direction of the velocity. The effect of the force, F, is to turn the body from the straight line PA, so that at the end of the time, dt, it is at a position, B. The effect of the force, F, is to move the body sideways a distance AB in the time, dt. If a is the acceleration produced by the force, F, in the direction AB, then

If F is measured in pounds, W is the weight of the body in pounds, and g is the acceleration of gravitation, the relation between F and a is given by

When the body arrives at B it is traveling in a direction slightly different from what it had at the point, P, but its speed is unchanged, because the deflecting force always acts normal to the direction of motion of the body and therefore has no tangential component. The path from P to B is curved. For such a short length as we are here considering the curve would be sensibly a circle and it is often convenient to know under such circumstances the size of this circle of curvature, or the radius of curvature. For any such case, where the circle is known to be tangent to the line, PA, at the point, P, and passes through the point, B, whose position is known with respect to P, the following rule, derived from plane geometry, and accurate only for differential quantities, is very convenient.

From this and equations 1 and 2 the ordinary expression for the acceleration of centripetal force is obtained:

In order to study each element in its simplest separate form let us next consider the effect of circular motion in the atmosphere upon barometer pressures as they would exist upon an earth without rotation. It will later be shown that the diurnal rotation of the earth has a decidedly important influence on the distribution of barometer pressures in a moving atmosphere, and it is simpler, and hence advantageous, to consider the earth at first without rotation.

Assume on some part of the earth's surface that over a large area the air is moving in an eddy or whirl in a counter-clockwise direction about some fixed center. This eddy might be 500 miles or more in diameter, with the wind having everywhere a velocity of 20 miles per hour. Such a circulation is called in meteorological parlance, a cyclone. If the air is at each point moving in a true circle around the center, this shows that at the center the barometric pressure must be lower than at the outskirts by the amount required by the formula for centripetal force. With a constant velocity, V, the barometric pressure along a radial line will change at a varying rate dependent upon the value of p. This rate or barometric gradient at the surface of the earth, expressed in inches of mercury difference in barometer reading per 100 miles of horizontal radial distance is

Barometric gradient = .25

V2 gp

=

V2

128p

(6)

in inches per 100 miles, in which V is the velocity of the wind in miles per hour, g is the acceleration of gravity in feet per second, and p is the radius in miles. For a velocity of 20 miles per hour, and a radius of 200 miles this gives a barometric gradient of .015 inch per 100

miles. This result shows that for ordinary conditions centripetal force has but small effect upon the distribution of barometric pressure, but for such extreme conditions as are reached in a West Indian hurricane the effect of rotation becomes very marked. In the above example with the circulation in a counter-clockwise direction the higher pressure would be on the right hand side of the moving air current.

An important relation in connection with a revolving cyclone such as that described above should be noted at this point. If such a whirling mass or disk of air should be, say, two miles thick and several hundred miles in diameter, the friction of the surface of the earth would be a disturbing element which would constantly tend to reduce the velocity of the moving air at the surface to a value less than that existing at higher altitudes above the earth's surface. This would upset any previously existing equilibrium between centrifugal force and barometric gradient. If at a height of 1000 feet, the atmosphere were revolving in circular paths with the horizontal barometric gradient just balancing the horizontal centrifugal force, then at lower elevations where the linear velocity of the air was less, the barometric gradient would more than suffice to overcome the centrifugal force, with the result that the lower air would be pushed in toward the center of the cyclone. Similarly at higher levels the barometric gradient would not suffice to overcome centrifugal force and the air would tend to move away from the center of rotation. The radial barometric gradient is so slight a quantity that these variations are perhaps not of prime importance, but observations in the air at different levels indicate a relative motion inward at the ground and an outward motion at high altitudes.

EFFECT OF THE EARTH'S ROTATION UPON FRICTIONLESS MOTION

First, let us consider the earth as a perfect sphere not in rotation. If a body could move on the earth's surface without any friction, then such a body once put into motion would move around the earth, under the force of gravity, pulling it constantly toward the earth's center, in a path which would be a great circle. For example, in figure 3, representing the earth, if a body were started at P on the equator in the direction PA it would move along the great circle represented by the line PABC until it passed entirely around the earth and came back to the point of beginning.

With such an earth not in rotation, the moving point, P, would actually traverse a great circle on the earth's surface and its velocity would be constant. Such a great circle might be said to correspond