Climate and Time in Their Geological Relations / A Theory of Secular Changes of the Earth's Climate

by James Croll

“(c) In imparting a downward movement to each new surface-stratum as its temperature undergoes reduction; so that the entire column may be said to be in a state of constant descent, like that which exists in the water of a tall jar when an opening is made at its bottom, and the water which flows away through it is replaced by an equivalent supply poured into the top of the jar” (§ 23).

But if this be his theory, as it evidently is, then the 4 foot-pounds (the amount of work performed by the descent of the water down the slope) comprehends all the work that gravitation can perform on a pound of water in making a complete circuit from the equator to the pole and from the pole back to the equator.

This, I trust, will be evident from the following considerations. When a pound of water has flowed down from the equator to the pole, it has descended 4 feet, and is then at the foot of the slope. Gravity has therefore no more power to pull it down to a lower level. It will not sink through the polar water, for it is not denser than the water beneath on which it rests. But it may be replied that although it will not sink through the polar water, it has nevertheless made the polar column heavier than the equatorial, and this excess of pressure forces a pound of water out from beneath and allows the column to descend. Suppose it may be argued that a quantity of water flows down from the equator, so as to raise the level of the polar water by, say, one foot. The polar column will now be rendered heavier than the equatorial by the weight of one foot of water. The pressure of the one foot will thus force a quantity of water laterally from the bottom and cause the entire column to descend till the level of equilibrium is restored. In other words, the polar column will sink one foot. Now in the sinking of this column work is performed by gravity. A certain amount of work is performed by gravity in causing the water to flow down the slope from the equator to the pole, and, in addition to this, a certain amount is performed by gravity in the vertical descent of the column.

I freely admit this to be sound reasoning, and admit that so much is due to the slope and so much to the vertical descent of the water. But here we come to the most important point, viz., is there the full slope of 4 feet and an additional vertical movement? Dr. Carpenter seems to conclude that there is, and that this vertical force is something in addition to the force which I derive from the slope. And here, I venture to think, is a radical error into which he has fallen in regard to the whole matter. Let it be observed that, when water circulates from difference of specific gravity, this vertical movement is just as real a part of the process as the flow down the slope; but the point which I maintain is that there is no additional power derived from this vertical movement over and above what is derived from the full slope—or, in other words, that this primum mobile, which he says I have overlooked, has in reality no existence.

Perhaps the following diagram will help to make the point still clearer:—

Fig. 1.

Let P (fig. 1) be the surface of the ocean at the pole, and E the surface at the equator; P O a column of water at the pole, and E Q a column at the equator. The two columns are of equal weight, and balance each other; but as the polar water is colder, and consequently denser than the equatorial, the polar column is shorter than the equatorial, the difference in the length of the two columns being 4 feet. The surface of the ocean at the equator E is 4 feet higher than the surface of the ocean at the pole P; there is therefore a slope of 4 feet from E to P. The molecules of water at E tend to flow down this slope towards P. The amount of work performed by gravity in the descent of a pound of water down this slope from E to P is therefore 4 foot-pounds.

But of course there can be no permanent circulation while the full slope remains. In order to have circulation the polar column must be heavier than the equatorial. But any addition to the weight of the polar column is at the expense of the slope. In proportion as the weight of the polar column increases the less becomes the slope. This, however, makes no difference in the amount of work performed by gravity.

Suppose now that water has flowed down till an addition of one foot of water is made to the polar column, and the difference of level, of course, diminished by one foot. The surface of the ocean in this case will now be represented by the dotted line P′ E, and the slope reduced from 4 feet to 3 feet. Let us then suppose a pound of water to leave E and flow down to P′; 3 foot-pounds will be the amount of work performed. The polar column being now too heavy by the extent of the mass of water P′ P one foot thick, its extra pressure causes a mass of water equal to P′ P to flow off laterally from the bottom of the column. The column therefore sinks down one foot till P′ reaches P. Now the pound of water in this vertical descent from P′ to P has one foot-pound of work performed on it by gravity; this added to the 3 foot-pounds derived from the slope, gives a total of 4 foot-pounds in passing from E to P′ and then from P′ to P. This is the same amount of work that would have been performed had it descended directly from E to P. In like manner it can be proved that 4 foot-pounds is the amount of work performed in the descent of every pound of water of the mass P′ P. The first pound which left E flowed down the slope directly to P, and performed 4 foot-pounds of work. The last pound flowed down the slope E P′, and performed only 3 foot-pounds; but in descending from P′ to P it performed the other one foot-pound. A pound leaving at a period exactly intermediate between the two flowed down 3½ feet of slope and descended vertically half a foot. Whatever path a pound of water might take, by the time that it reached P, 4 foot-pounds of work would be performed. But no further work can be performed after it reaches P.