Dynamically, the tornado may be treated as a rotating pillar of air in which each mass of fluid fairly retains its angular momentum. This means that for any mass of the whirling air the radius of its path, multiplied by its circular speed, remains a constant product; in other words, the velocity of whirl varies inversely as the radius. Accordingly, the circular velocity is exceedingly rapid where the radius is very small. Now, when any mass runs round a circle its centrifugal force is known to be directly as the square of the speed of its centroid and inversely as the radius. But by the above assumption the speed itself is inversely as the radius. Hence, the centrifugal force varies inversely as the cube of the radius of the inflowing mass of air. This centrifugal force, acting on the inner layers of air of the rotating column, must be supported by the pressure against them exerted by the outer layers as they pass inward. Thus there is a strong barometric gradient from the remote still air toward the swiftly whirling parts of the vortex.
It follows from the above argument that inside a tornado the barometric pressure may be much below the normal; and it is easy to see that if a barometer, starting from some point on the tornado base, be moved vertically upward it must show a declining pressure, but if moved upward and outward it may be made to show a constant pressure all the way to the upper portion of the vortex. The instrument would thus travel along an isobaric, bell-shaped surface opening upward. On a series, therefore, of concentric circles on the base of a tornado, we may erect a family of coaxial bell-shaped surfaces to mark the points of equal pressure, and thus map out the isobars of the vortex. Inside these coaxial surfaces reaching to earth, others of still lower pressure may be drawn tapering downward to a rounded point and terminating at various places on the axis. In an actual tornado one of these infinitely numerous funnel-shaped isobaric surfaces may become distinctly outlined and visible, if the air has sufficient moisture to start precipitation when it reaches a surface of suitably low pressure. This quite usually occurs in Nature, the funnel sometimes reaching to earth, sometimes only part way, according to the pressure at which precipitation begins, this pressure depending, of course, on the percentage of humidity of the uprushing air.
The form of the funnel-like cloud ere it reaches the earth is interesting. Being an isobaric surface, it would support in static equilibrium a free particle resting on it and sharing its rotatory motion. The lower rounded part of the funnel is parabolic, the upper outer part hyperbolic; the two together delineating the well-known Rankine double vortex of hydrodynamics. Students of hydrostatics know that when a glass of water is spun round its axis at a fixed velocity, the dimple observed is of parabolic form, and if frozen will sustain in repose a small shot resting on its surface and whirling with it. Similarly the lower part of the funnel is parabolic because in it the air rotates, as one solid body, while the broader part of the funnel is hyperbolic because in it the air has a speed inversely proportional to its radius of motion.
If everywhere in a tornado the circular velocity of the inflowing air were inversely proportional to the radius, as above assumed, the speed near the axis would be indefinitely great. This cannot be admitted. Practically, the inflow ceases when the centrifugal force of the gyrating stratum equals the pressure urging it toward the axis. Within this stratum is a column of air rotating everywhere with constant angular velocity about the vortical axis, and thus having quite calm air at its center. Outside this solidly rotating core the air spirals radially inward and upward. Some idea of the stream lines in such spiral flow may be obtained from Fig. 50 if a rapid circular motion be added to the inward and upward velocity represented by the arrows.
In the foregoing discussion no account of friction was taken. Near the earth’s surface this dampens the whirl and centrifugal force, so that the air flows more directly into the vortex, while farther aloft the centrifugal force near the axis so effectually checks the inflow as to allow the central core of air to rush up nearly unimpeded, as in a walled flue, taking its draught mostly from the lower part. As a consequence, the upward speed of the heated air in the tornado tube may be enormous, supporting in its stream objects of considerable mass.
Morey
Fig. 50.—Funnel-like Cloud
Sometimes Observed in a Tornado.
The true horizontal speed anywhere in a tornado is compounded of the velocities of gyration and of translation, as in the cyclone. Hence the advancing side may be considerably the swifter and more destructive, particularly more destructive since the impact of air increases as the square of the velocity. If the vortex were stationary it would be equally dangerous on all sides, standing erect and symmetrical; but it drifts with the whole mass of air, sometimes quite swiftly and often with varying speed of travel at different levels; thus, in its slenderest forms, appearing bent and not infrequently twisted, as it advances writhing serpentlike through the sky. Furthermore, the intensity of whirl may fluctuate momentarily, with consequent shifting of the isobaric surface, including that one whose form is visible by reason of incipient condensation; and thus the funnel-like misty tongue appears to dart earthward as a foggy downshoot from the cloud above, whereas its parts are really rushing upward at all times very swiftly, whether visible or not. This agile protrusion of the nimbus, now a tongue, now a dark and mighty tower, is the strenuous part of the storm, the abominated “twister” which the Kansan farmer sedulously shuns, or peeps at from a hole in the ground. Unwelcome, indeed, are its visitations, when, with mickle and multitudinous roar, it claps his house in sudden darkness, hurls it aloft and sows its sacred relics over all the adjoining township, “that with the hurly-burly hell itself awakes.”
Theory, as well as experience, accredits the tornado with vast energy and power. For, suppose a surface stratum of air one mile in area and one thousand feet thick to increase in absolute temperature one per cent, thus uplifting the superincumbent atmosphere ten feet. The total energy stored in this way equals the weight lifted multiplied by its upward displacement. The weight is a ton per square foot and the displacement is ten feet; hence the stored energy is ten-foot tons per square foot of the heated tract, or about 280,000,000 foot tons for the square mile of heated air. This is equivalent to the work of one million horses for over a quarter of an hour. A goodly percentage of this stored work may be converted into kinetic energy in the active part of the dry tornado. It is the energy of a vast reservoir suddenly gushing through a tall penstock. It is a colossal upward cataract, an aërial Niagara, a Johnstown flood suddenly liberated and quickly spent.