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

by James Croll

The slope E P being 4 feet, the slope E′ P′ is consequently 2 feet; the mean slope for the entire mass is therefore 3 feet. The mean amount of work performed by the descent of the mass will of course be 3 foot-pounds per pound of water. The amount of work performed by the vertical descent of P′ P ought therefore to be one foot-pound per pound. That this is the amount will be evident thus:—The transference of the one foot of water from the equatorial column to the polar disturbs the equilibrium by making the equatorial column too light by one foot of water and the polar column too heavy by the same amount of water. The polar column will therefore tend to sink, and the equatorial to rise till equilibrium is restored. The difference of weight of the two columns being equal to 2 feet of water, the polar column will begin to descend with a pressure of 2 feet of water; and the equatorial column will begin to rise with an equal amount of pressure. When the polar column has descended half a foot the equatorial column will have risen half a foot. The pressure of the descending polar column will now be reduced to one foot of water. And when the polar column has descended another foot, P′ will have reached P, and E′ will have reached E; the two columns will then be in equilibrium. It therefore follows that the mean pressure with which the polar column descended the one foot was equal to the pressure of one foot of water. Consequently the mean amount of work performed by the descent of the mass was equal to one foot-pound per pound of water; this, added to the 3 foot-pounds derived from the slope, gives a total of 4 foot-pounds.

In whatever way we view the question, we are led to the conclusion that if 4 feet represent the amount of slope between the equatorial and polar columns when the two are in equilibrium, then 4 foot-pounds is the total amount of work that gravity can perform upon a pound of water in overcoming the resistance to motion in its passage from the equator to the pole down the slope, and then in its vertical descent to the bottom of the ocean.

But it will be replied, not only does the one foot of water P′ P descend, but the entire column P O, 10,000 feet in length, descends also. What, then, it will be asked, becomes of the force which gravity exerts in the descent of this column? We shall shortly see that this force is entirely applied in work against gravity in other parts of the circuit; so that not a single foot-pound of this force goes to overcome cohesion, friction, and other resistances; it is all spent in counteracting the efforts which gravity exerts to stop the current in another part of the circuit.

I shall now consider the next part of the movement, viz., the under or return current from the bottom of the polar to the bottom of the equatorial column. What produces this current? It is needless to say that it cannot be caused directly by gravity. Gravitation cannot directly draw any body horizontally along the earth’s surface. The water that forms this current is pressed out laterally by the weight of the polar column, and flows, or rather is pushed, towards the equator to supply the vacancy caused by the ascent of the equatorial column. There is a constant flow of water from the equator to the poles along the surface, and this draining of the water from the equator is supplied by the under or return current from the poles. But the only power which can impel the water from the bottom of the polar column to the bottom of the equatorial column is the pressure of the polar column. But whence does the polar column derive its pressure? It can only press to the extent that its weight exceeds that of the equatorial column. That which exerts the pressure is therefore the mass of water which has flowed down the slope from the equator upon the polar column. It is in this case the vertical movement that causes this under current. The energy which produces this current must consequently be derived from the 4 foot-pounds resulting from the slope; for the energy of the vertical movement, as has already been proved, is derived from this source; or, in other words, whatever power this vertical movement may exert is so much deducted from the 4 foot-pounds derived from the full slope.

Let us now consider the fourth and last movement, viz., the ascent of the under current to the surface of the ocean at the equator. When this cold under current reaches the equatorial regions, it ascends to the surface to the point whence it originally started on its circuit. What, then, lifts the water from the bottom of the equatorial column to its top? This cannot be done directly, either by heat or by gravity. When heat, for example, is applied to the bottom of a vessel, the heated water at the bottom expands and, becoming lighter than the water above, rises through it to the surface; but if the heat be applied to the surface of the water instead of to the bottom, the heat will not produce an ascending current. It will tend rather to prevent such a current than to produce one—the reason being that each successive layer of water will, on account of the heat applied, become hotter and consequently lighter than the layer below it, and colder and consequently heavier than the layer above it. It therefore cannot ascend, because it is too heavy; nor can it descend, because it is too light. But the sea in equatorial regions is heated from above, and not from below; consequently the water at the bottom does not rise to the surface at the equator in virtue of any heat which it receives. A layer of water can never raise the temperature of a layer below it to a higher temperature than itself; and since it cannot do this, it cannot make the layer under it lighter than itself. That which raises the water at the equator, according to Dr. Carpenter’s theory, must be the downward pressure of the polar column. When water flows down the slope from the equator to the pole, the polar column, as we have seen, becomes too heavy and the equatorial column too light; the former then sinks and the latter rises. It is the sinking of the polar column which raises the equatorial one. When the polar column descends, as much water is pressed in underneath the equatorial column as is pressed from underneath the polar column. If one foot of water is pressed from under the polar column, a foot of water is pressed in under the equatorial column. Thus, when the polar column sinks a foot, the equatorial column rises to the same extent. The equatorial water continuing to flow down the slope, the polar column descends: a foot of water is again pressed from underneath the polar column and a foot pressed in under the equatorial. As foot after foot is thus removed from the bottom of the polar column while it sinks, foot after foot is pushed in under the equatorial column while it rises; so by this means the water at the surface of the ocean in polar regions descends to the bottom, and the water at the bottom in equatorial regions ascends to the surface—the effect of solar heat and polar cold continuing, of course, to maintain the surface of the ocean in equatorial regions at a higher level than at the poles, and thus keeping up a constant state of disturbed equilibrium. Or, to state the matter in Dr. Carpenter’s own words, “The cold and dense polar water, as it flows in at the bottom of the equatorial column, will not directly take the place of that which has been drafted off from the surface; but this place will be filled by the rising of the whole superincumbent column, which, being warmer, is also lighter than the cold stratum beneath. Every new arrival from the poles will take its place below that which precedes it, since its temperature will have been less affected by contact with the warmer water above it. In this way an ascending movement will be imparted to the whole equatorial column, and in due course every portion of it will come under the influence of the surface-heat of the sun.”[73]

But the agency which raises up the water of the under current to the surface is the pressure of the polar column. The equatorial column cannot rise directly by means of gravity. Gravity, instead of raising the column, exerts all its powers to prevent its rising. Gravity here is a force acting against the current. It is the descent of the polar column, as has been stated, that raises the equatorial column. Consequently the entire amount of work performed by gravity in pulling down the polar column is spent in raising the equatorial column. Gravity performs exactly as much work in preventing motion in the equatorial column as it performs in producing motion in the polar column; so that, so far as the vertical parts of Dr. Carpenter’s circulation are concerned, gravity may be said neither to produce motion nor to prevent it. And this remark, be it observed, applies not only to P O and E Q, but also to the parts P′ P and E E′ of the two columns. When a mass of water E E′, say one foot deep, is removed off the equatorial column and placed upon the polar column, the latter column is then heavier than the former by the weight of two feet of water. Gravity then exerts more force in pulling the polar column down than it does in preventing the equatorial column from rising; and the consequence is that the polar column begins to descend and the equatorial column to rise. But as the polar column continues to descend and the equatorial to rise, the power of gravity to produce motion in the polar column diminishes, and the power of gravity to prevent motion in the equatorial column increases; and when P′ descends to P and E′ rises to E, the power of gravity to prevent motion in the equatorial column is exactly equal to the power of gravity to produce motion in the polar column, and consequently motion ceases. It therefore follows that the entire amount of work performed by the descent of P′ P is spent in raising E′ E against gravity.

It follows also that inequalities in the sea-bottom cannot in any way aid the circulation; for although the cold under current should in its progress come to a deep trough filled with water less dense than itself, it would no doubt sink to the bottom of the hollow; yet before it could get out again as much work would have to be performed against gravity as was performed by gravity in sinking it. But whilst inequalities in the bed of the ocean would not aid the current, they would nevertheless very considerably retard it by the obstructions which they would offer to the motion of the water.

We have been assuming that the weight of P′ P is equal to that of E E′; but the mass P′ P must be greater than E E′ because P′ P has not only to raise E E′, but to impel the under current—to push the water along the sea-bottom from the pole to the equator. So we must have a mass of water, in addition to P′ P, placed on the polar column to enable it to produce the under current in addition to the raising of the equatorial column.

It follows also that the amount of work which can be performed by gravity depends entirely on the difference of temperature between the equatorial and the polar waters, and is wholly independent of the way in which the temperature may decrease from the equator to the poles. Suppose, in agreement with Dr. Carpenter’s idea,[74] that the equatorial heat and polar cold should be confined to limited areas, and that through the intermediate space no great difference of temperature should prevail. Such an arrangement as this would not increase the amount of work which gravity could perform; it would simply make the slope steeper at the two extremes and flatter in the intervening space. It would no doubt aid the surface-flow of the water near the equator and the poles, but it would retard in a corresponding degree the flow of the water in the intermediate regions. In short, it would merely destroy the uniformity of the slope without aiding in the least degree the general motion of the water.

It is therefore demonstrable that the energy derived from the full slope, whatever that slope may be, comprehends all that can possibly be obtained from gravity.