CHAPTER IX.

EXAMINATION OF THE GRAVITATION THEORY OF OCEANIC CIRCULATION.—THE MECHANICS OF DR. CARPENTER’S THEORY.

Experimental Illustration of the Theory.—The Force exerted by Gravity.—Work performed by Gravity.—Circulation not by Convection.—Circulation depends on Difference in Density of the Equatorial and Polar Columns.—Absolute Amount of Work which can be performed by Gravity.—How Underflow is produced.—How Vertical Descent at the Poles and Ascent at the Equator is produced.—The Gibraltar Current.—Mistake in Mechanics concerning it.—The Baltic Current.

_Experiment to illustrate Theory._—In support of the theory of a general movement of water between equatorial and polar regions, Dr. Carpenter adduces the authority of Humboldt and of Prof. Buff.[69] I have been unable to find anything in the writings of either from which it can be inferred that they have given this matter special consideration. Humboldt merely alludes to the theory, and that in the most casual manner; and that Prof. Buff has not carefully investigated the subject is apparent from the very illustration quoted by Dr. Carpenter from the “Physics of the Earth.” “The water of the ocean at great depths,” says Prof. Buff, “has a temperature, even under the equator, nearly approaching to the freezing-point. This low temperature cannot depend on any influence of the sea-bottom.... The fact, however, is explained by a continual current of cold water flowing from the polar regions towards the equator. The following well-known experiment clearly illustrates the manner of this movement. A glass vessel is to be filled with water with which some powder has been mixed, and is then to be heated _at bottom_. It will soon be seen, from the motion of the particles of powder, that currents are set up in opposite directions through the water. Warm water rises from the bottom up through the middle of the vessel, and spreads over the surface, while the colder and therefore heavier liquid falls down at the sides of the glass.”

This illustration is evidently intended to show not merely the form and direction of the great system of oceanic circulation, but also the mode in which the circulation is induced by heat. It is no doubt true that if we apply heat (say that of a spirit-lamp) to the bottom of a vessel filled with water, the water at the bottom of the vessel will become heated and rise to the surface; and if the heat be continued an ascending current of warm water will be generated; and this, of course, will give rise to a compensating under current of colder water from all sides. In like manner it is also true that, if heat were applied to the bottom of the ocean in equatorial regions, an ascending current of hot water would be also generated, giving rise to an under current of cold water from the polar regions. But all this is the diametrically opposite of what actually takes place in nature. The heat is not applied to the bottom of the ocean, so as to make the water there lighter than the water at the surface, and thus to generate an ascending current; but the heat is applied to the surface of the ocean, and the effect of this is to prevent an ascending current rather than to produce one, for it tends to keep the water at the surface lighter than the water at the bottom. In order to show how the heat of the sun produces currents in the ocean, Prof. Buff should have applied the heat, not to the bottom of his vessel, but to the upper surface of the water. But this is not all, the form of the vessel has something to do with the matter. The wider we make the vessel in proportion to its depth, the more difficult it is to produce currents by means of heat. But in order to represent what takes place in nature, we ought to have the same proportion between the depth and the superficial area of the water in our vessel as there is between the depth and the superficial area of the sea. The mean depth of the sea may be taken roughly to be about three miles.[70] The distance between pole and pole we shall take in round numbers to be 12,000 miles. The sun may therefore be regarded as shining upon a circular sea 12,000 miles in diameter and three miles deep. The depth of the sea to its diameter is therefore as 1 to 4,000. Suppose, now, that in our experiment we make the depth of our vessel one inch, we shall require to make its diameter 4,000 inches, or 333 feet, say, in round numbers, 100 yards in diameter. Let us, then, take a pool of water 100 yards in diameter, and one inch deep. Suppose the water to be at 32°. Apply heat to the upper surface of the pool, so as to raise the temperature of the surface of the water to 80° at the centre of the pool, the temperature diminishing towards the edge, where it is at 32°. It is found that at a depth of two miles the temperature of the water at the equator is about as low as that of the poles. We must therefore suppose the water at the centre of our pool to diminish in temperature from the surface downwards, so that at a depth of half an inch the water is at 32°. We have in this case a thin layer of warm water half an inch thick at the centre, and gradually thinning off to nothing at the edge of the pool. The lightest water, be it observed, is at the surface, so that an ascending or a descending current is impossible. The only way whereby the heat applied can have any tendency to produce motion is this:—The heating of the water expands it, consequently the surface of the pool must stand at a little higher level at its centre than at its edge, where no expansion takes place; and therefore, in order to restore the level of the pool, the water at the centre will tend to flow towards the sides. But what is the amount of this tendency? Its amount will depend upon the amount of slope, but the slope in the case under consideration amounts to only 1 in 7,340,000.

_Dr. Carpenter’s Experiment._—In order to obviate the objection to Professor Buff’s experiment Dr. Carpenter has devised another mode. But I presume his experiment was intended rather to illustrate the way in which the circulation of the ocean, according to his theory, takes place, than to prove that it actually does take place. At any rate, all that can be claimed for the experiment is the proof that water will circulate in consequence of difference of specific gravity resulting from difference of temperature. But this does not require proof, for no physicist denies it. The point which requires to be proved is this. Is the difference of specific gravity which exists in the ocean sufficient to produce the supposed circulation? Now his mode of experimenting will not prove this, unless he makes his experiment agree with the conditions already stated.

But I decidedly object to the water being heated in the way in which it has been done by him in his experiment before the Royal Geographical Society; for I feel somewhat confident that in this experiment the circulation resulted not from difference of specific gravity, as was supposed, but rather from the way in which the heat was applied. In that experiment the one half of a thick metallic plate was placed in contact with the upper surface of the water at one end of the trough; the other half, projecting over the end of the trough, was heated by means of a spirit-lamp. It is perfectly obvious that though the temperature of the great mass of the water under the plate might not be raised over 80° or so, yet the molecules in contact with the metal would have a very high temperature. These molecules, in consequence of their expansion, would be unable to sink into the cooler and denser water underneath, and thus escape the heat which was being constantly communicated to them from the heated plate. But escape they must, or their temperature would continue to rise until they would ultimately burst into vapour. They cannot ascend, neither can they descend: they therefore must be expelled by the heat from the plate in a horizontal direction. The next layer of molecules from beneath would take their place and would be expelled in a similar manner, and this process would continue so long as the heat was applied to the plate. A circulation would thus be established by the direct expansive force of vapour, and not in any way due to difference of specific gravity, as Dr. Carpenter supposes.

But supposing the heated bar to be replaced by a piece of ice, circulation would no doubt take place; but this proves nothing more than that difference of density will produce circulation, which is what no one calls in question.

The case referred to by Dr. Carpenter of the heating apparatus in London University is also unsatisfactory. The water leaves the boiler at 120° and returns to it at 80°. The difference of specific gravity between the water leaving the boiler and the water returning to it is supposed to produce the circulation. It seems to me that this difference of specific gravity has nothing whatever to do with the matter. The cause of the circulation must be sought for in the boiler itself, and not in the pipes. The heat is applied to the bottom of the boiler, not to the top. What is the temperature of the molecules in contact with the bottom of the boiler directly over the fire, is a question which must be considered before we can arrive at a just determination of the causes which produce circulation in the pipes of a heating apparatus such as that to which Dr. Carpenter refers. But, in addition to this, as the heat is applied to the bottom of the boiler and not to the top, convection comes into play, a cause which, as we shall find, does not come into play in the theory of oceanic circulation at present under our consideration.

_The Force exerted by Gravity._—Dr. Carpenter speaks of his doctrine of a general oceanic circulation sustained by difference of temperature alone, “as one of which physical geographers could not recognise the importance, so long as they remained under the dominant idea that the temperature of the deep sea is everywhere 39°.” And he affirms that “until it is clearly apprehended that sea-water becomes more and more dense as its temperature is reduced, the immense motive power of polar cold cannot be understood.” But in chap. vii. and also in the Phil. Mag. for October, 1870 and 1871, I proved that if we take 39° as the temperature of maximum density the force exerted by gravity tending to produce circulation is just as great as when we take 32°. The reason for this is that when we take 32° as the temperature of maximum density, although we have, it is true, a greater elevation of the ocean above the place of maximum density, yet this latter occurs at the poles; while on the other hand, when we take 39°, the difference of level is less—the place not being at the poles but in about lat. 56°. Now the shorter slope from the equator to lat. 56° is as steep as the larger one from the equator to the poles, and consequently gravity exerts as much force in the production of motion in the one case as in the other. Sir John Herschel, taking 39° as the temperature of maximum density, estimated the slope at 1/32nd of an inch per mile, whereas we, taking 32° as the actual temperature of maximum density of the polar seas and calculating from modern data, find that the slope is not one-half that amount, and that the force of gravity tending to produce circulation is much less than Herschel concluded it to be. The reason, therefore, why physical geographers did not adopt the theory that oceanic circulation is the result of difference of temperature could not possibly be the one assigned by Dr. Carpenter, viz., that they had under-estimated the force of gravity by taking 39° instead of 32° as the temperature of maximum density.

_The Work performed by Gravity._—But in order clearly to understand this point, it will be better to treat the matter according to the third method, and consider not the mere _force_ of gravity impelling the waters, but the amount of _work_ which gravitation is capable of performing.

Let us then assume the correctness of my estimate, that the height of the surface of the ocean at the equator above that at the poles is 4 feet 6 inches, for in representing the mode in which difference of specific gravity produces circulation it is of no importance what we may fix upon as the amount of the slope. In order, therefore, to avoid fractions of a foot, I shall take the slope at 4 feet instead of 4½ feet, which it actually is. A pound of water in flowing down this slope from the equator to either of the poles will perform 4 foot-pounds of work; or, more properly speaking, gravitation will. Now it is evident that when this pound of water has reached the pole, it is at the bottom of the slope, and consequently cannot descend further. Gravity, therefore, cannot perform any more work upon it; as it can only do so while the thing acted upon continues to descend—that is, moves under the force exerted. But the water will not move under the influence of gravity unless it move downward; it being in this direction only that gravity acts on the water. “But,” says Dr. Carpenter, “the effect of surface-cold upon the water of the polar basin will be to reduce the temperature of its whole mass below the freezing-point of fresh water, the surface-stratum _sinking_ as it is cooled in virtue of its diminished bulk and increased density, and being replaced by water not yet cooled to the same degree.”[71] By the cooling of the whole mass of polar water by cold and the heating of the water at the equator by the sun’s rays the polar column of water, as we have seen, is rendered denser than the equatorial one, and in order that the two may balance each other, the polar column is necessarily shorter than the equatorial by 4 feet; and thus it is that the slope of 4 feet is formed. It is perfectly true that the water which leaves the equator warm and light, becomes by the time it reaches the pole cold and dense. But unless it be denser than the underlying polar water it will not sink down _through_ it.[72] We are not told, however, why it should be colder than the whole mass underneath, which, according to Dr. Carpenter, is cooled by polar cold. But that he does suppose it to sink to the bottom in consequence of its contraction by cold would appear from the following quotation:—

“Until it is clearly apprehended that sea-water becomes more and more dense as its temperature is reduced, and that it consequently continues to sink until it freezes, the immense motor power of polar cold cannot be apprehended. But when this has been clearly recognised, it is seen that the application of _cold at the surface_ is precisely equivalent as a moving power to that application of _heat at the bottom_ by which the circulation of water is sustained in every heating apparatus that makes use of it” (§ 25).

The application of cold at the surface is thus held to be equivalent as a motor power to the application of heat at the bottom. But heat applied to the bottom of a vessel produces circulation by _convection_. It makes the molecules at the bottom expand, and they, in consequence of buoyancy, rise _through_ the water in the vessel. Consequently if the action of cold at the surface in polar regions is equivalent to that of heat, the cold must contract the molecules at the surface and make them sink _through_ the mass of polar water beneath. But assuming this to be the meaning in the passage just quoted, how much colder is the surface water than the water beneath? Let us suppose the difference to be one degree. How much work, then, will gravity perform upon this one pound of water which is one degree colder than the mass beneath supposed to be at 32°? The force with which the pound of water will sink will not be proportional to its weight, but to the difference of weight between it and a similar bulk of the water through which it sinks. The difference between the weight of a pound of water at 31° and an equal volume of water at 32° is 1/29,000th of a pound. Now this pound of water in sinking to a depth of 10,000 feet, which is about the depth at which a polar temperature is found at the equator, would perform only one-third of a foot-pound of work. And supposing it were three degrees colder than the water beneath, it would in sinking perform only one foot-pound. This would give us only 4 + 1 = 5 foot-pounds as the total amount that could be performed by gravitation on the pound of water from the time that it left the equator till it returned to the point from which it started. The amount of work performed in descending the slope from the equator to the pole and in sinking to a depth of 10,000 feet or so through the polar water assumed to be warmer than the surface water, comprehends the total amount of work that gravitation can possibly perform; so that the amount of force gained by such a supposition over and above that derived from the slope is trifling.

It would appear, however, that this is not what is meant after all. What Dr. Carpenter apparently means is this: when a quantity of water, say a layer one foot thick, flows down from the equator to the pole, the polar column becomes then heavier than the equatorial by the weight of this additional layer. A layer of water equal in quantity is therefore pressed away from the bottom of the column and flows off in the direction of the equator as an under current, the polar column at the same time sinking down one foot until equilibrium of the polar and equatorial columns is restored. Another foot of water now flows down upon the polar column and another foot of water is displaced from below, causing, of course, the column to descend an additional foot. The same process being continually repeated, a constant downward motion of the polar column is the result. Or, perhaps, to express the matter more accurately, owing to the constant flow of water from the equatorial regions down the slope, the weight of the polar column is kept always in excess of that of the equatorial; therefore the polar column in the effort to restore equilibrium is kept in a constant state of descent. Hence he terms it a “vertical” circulation. The following will show Dr. Carpenter’s theory in his own words:—

“The action of cold on the surface water of each polar area will be exerted as follows:—

“(_a_) In diminishing the height of the polar column as compared with that of the equatorial, so that a lowering of its _level_ is produced, which can only be made good by a surface-flow from the latter towards the former.

“(_b_) In producing an excess in the downward _pressure_ of the column when this inflow has restored its level, in virtue of the increase of specific gravity it has gained by its reduction in volume; whereby a portion of its heavy bottom-water is displaced laterally, causing a further reduction of level, which draws in a further supply of the warmer and lighter water flowing towards its surface.

“(_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:—

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.

But some will ask, in regard to the vertical movement, is it only in the descent of the water from P′ to P that work is performed? Water cannot descend from P′ to P, it will be urged, unless the entire column P O underneath descend also. But the column P O descends by means of gravity. Why, then, it will be asked, is not the descent of the column a motive power as real as the descent of the mass of water P′ P?

That neither force nor energy can be derived from the mere descent of the polar column P O is demonstrable thus:—The reason why the column P O descends is because, in consequence of the mass of water P′ P resting on it, its weight is in excess of the equatorial column E Q. But the force with which the column descends is equal, not to the weight of the column, but to the weight of the mass P′ P; consequently as much work would be performed by gravity in the descent of the mass P′ P (the one foot of water) alone as in the descent of the entire column P′ O, 10,000 feet in height. Suppose a ton weight is placed in each scale of a balance: the two scales balance each other. Place a pound weight in one of the scales along with the ton weight and the scale will descend. But it descends, not with the pressure of a ton and a pound, but with the pressure of the pound weight only. In the descent of the scale, say, one foot, gravity can perform only one foot-pound of work. In like manner, in the descent of the polar column, the only work available is the work of the mass P′ P laid on the top of the column. But it must be observed that in the descent of the column from P′ to P, a distance of one foot, each pound of water of the mass P′ P does not perform one foot-pound of work; for the moment that a molecule of water reaches P, it then ceases to perform further work. The molecules at the surface P′ descend one foot before reaching P; the molecules midway between P′ and P descend only half a foot before reaching P, and the molecules at the bottom of the mass are already at P, and therefore cannot perform any work. The mean distance through which the entire mass performs work is therefore half a foot. One foot-pound per pound of water represents in this case the amount of work derived from the vertical movement.

That such is the case is further evident from the following considerations. Before the polar column begins to descend, it is heavier than the equatorial by the weight of one foot of water; but when the column has descended half a foot, the polar column is heavier than the equatorial by the weight of only half a foot of water; and, as the column continues to descend, the force with which it descends continues to diminish, and when it has sunk to P the force is zero. Consequently the mean pressure or weight with which the one foot of water P′ P descended was equal to that of a layer of half a foot of water; in other words, each pound of water, taking the mass as a whole, descended with the pressure or weight of half a pound. But a half pound descending one foot performs half a foot-pound; so that whether we consider the _full pressure acting through the mean distance, or the mean pressure acting through the full distance, we get the same result_, viz. a half foot-pound as the work of vertical descent.

Now it will be found, as we shall presently see, that if we calculate the mean amount of work performed in descending the slope from the equator to the pole, 3½ foot-pounds per pound of water is the amount. The water at the bottom of the mass P P′ moved, of course, down the full slope E P 4 feet. The water at the top of the mass which descended from E to P′ descended a slope of only 3 feet. The mean descent of the whole mass is therefore 3½ feet. And this gives 3½ foot-pounds as the mean amount of work per pound of water in descending the slope; this, added to the half foot-pound derived from vertical descent, gives 4 foot-pounds as the total amount of work per pound of the mass.

I have in the above reasoning supposed one foot of water accumulated on the polar column before any vertical descent takes place. It is needless to remark that the same conclusion would have been arrived at, viz., that the total amount of work performed is 4 foot-pounds per pound of water, supposing we had considered 2 feet, or 3 feet, or even 4 feet of water to have accumulated on the polar column before vertical motion took place.

I have also, in agreement with Dr. Carpenter’s mode of representing the operation, been considering the two effects, viz., the flowing of the water down the slope and the vertical descent of the polar column as taking place alternately. In nature, however, the two effects take place simultaneously; but it is needless to add that the amount of work performed would be the same whether the effects took place alternately or simultaneously.

I have also represented the level of the ocean at the equator as remaining permanent while the alterations of level were taking place at the pole. But in representing the operation as it would actually take place in nature, we should consider the equatorial column to be lowered as the polar one is being raised. We should, for example, consider the one foot of water P′ P put upon the polar column as so much taken off the equatorial column. But in viewing the problem thus we arrive at exactly the same results as before.

Let P (Fig. 2), as in Fig. 1, be the surface of the ocean at the pole, and E the surface at the equator, there being a slope of 4 feet from E to P. Suppose now a quantity of water, E E′, say, one foot thick, to flow from off the equatorial regions down upon the polar. It will thus lower the level of the equatorial column by one foot, and raise the level of the polar column by the same amount. I may, however, observe that the one foot of water in passing from E to P would have its temperature reduced from 80° to 32°, and this would produce a slight contraction. But as the weight of the mass would not be affected, in order to simplify our reasoning we may leave this contraction out of consideration. Any one can easily satisfy himself that the assumption that E E′ is equal to P′ P does not in any way affect the question at issue—the only effect of the contraction being to _increase_ by an infinitesimal amount the work done in descending the slope, and to _diminish_ by an equally infinitesimal amount the work done in the vertical descent. If, for example, 3 foot-pounds represent the amount of work performed in descending the slope, and one foot-pound the amount performed in the vertical descent, on the supposition that E′ E does not contract in passing to the pole, then 3·0024 foot-pounds will represent the work of the slope, and 0·9976 foot-pounds the work of vertical descent when allowance is made for the contraction. But the total amount of work performed is the same in both cases. Consequently, to simplify our reasoning, we may be allowed to assume P′ P to be equal to E E′.

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_.

It cannot be urged as an objection to what has been advanced that I have determined simply the amount of the force acting on the water at the surface of the ocean and not that on the water at all depths—that I have estimated the amount of work which gravity can perform on a given quantity of water at the surface, but not the total amount of work which gravity can perform on the entire ocean. This objection will not stand, because it is at the surface of the ocean where the greatest difference of temperature, and consequently of density, exists between the equatorial and polar waters, and therefore there that gravity exerts its greatest force. And if gravity be unable to move the water at the surface, it is much less able to do so under the surface. So far as the question at issue is concerned, any calculations as to the amount of force exerted by gravity at various depths are needless.

It is maintained also that the winds cannot produce a vertical current except under some very peculiar conditions. We have already seen that, according to Dr. Carpenter’s theory, the vertical motion is caused by the water flowing off the equatorial column, down the slope, upon the polar column, thus destroying the equilibrium between the two by diminishing the weight of the equatorial column and increasing that of the polar column. In order that equilibrium may be restored, the polar column sinks and the equatorial one rises. Now must not the same effect occur, supposing the water to be transferred from the one column to the other, by the influence of the winds instead of by the influence of gravity? The vertical descent and ascent of these columns depend entirely upon the difference in their weights, and not upon the nature of the agency which makes this difference. So far as difference of weight is concerned, 2 feet of water, propelled down the slope from the equatorial column to the polar by the winds, will produce just the same effect as though it had been propelled by gravity. If vertical motion follows as a necessary consequence from a transference of water from the equator to the poles by gravity, it follows equally as a necessary consequence from the same transference by the winds; so that one is not at liberty to advocate a vertical circulation in the one case and to deny it in the other.

_Gravitation Theory of the Gibraltar Current._—If difference of specific gravity fails to account for the currents of the ocean in general, it certainly fails in a still more decided manner to account for the Gibraltar current. The existence of the submarine ridge between Capes Trafalgar and Spartel, as was shown in the Phil. Mag. for October, 1871, p. 269, affects currents resulting from difference of specific gravity in a manner which does not seem to have suggested itself to Dr. Carpenter. The pressure of water and other fluids is not like that of a solid—not like that of the weight in the scale of a balance, simply a downward pressure. Fluids press downwards like the solids, but they also press laterally. The pressure of water is hydrostatic. If we fill a basin with water or any other fluid, the fluid remains in perfect equilibrium, provided the sides of the basin be sufficiently strong to resist the pressure. The Mediterranean and Atlantic, up to the level of the submarine ridge referred to, may be regarded as huge basins, the sides of which are sufficiently strong to resist all pressure. It follows that, however much denser the water of the Mediterranean may be than that of the Atlantic, it is only the water above the level of the ridge that can possibly exercise any influence in the way of disturbing equilibrium, so as to cause the level of the Mediterranean to stand lower than that of the Atlantic. The water of the Atlantic below the level of this ridge might be as light as air, and that of the Mediterranean as heavy as molten lead, but this could produce no disturbance of equilibrium; and if there be no difference of density between the Atlantic and the Mediterranean waters from the surface down to the level of the top of the ridge, then there can be nothing to produce the circulation which Dr. Carpenter infers. Suppose both basins empty, and dense water to be poured into the Mediterranean, and water less dense into the Atlantic, until they are both filled up to the level of the ridge, it is evident that the heavier water in the one basin can exercise no influence in raising the level of the lighter water in the other basin, the entire pressure being borne by the sides of the basins. But if we continue to pour in water till the surface is raised, say one foot, above the level of the ridge, then there is nothing to resist the lateral pressure of this one foot of water in the Mediterranean but the counter pressure of the one foot in the Atlantic. But as the Mediterranean water is denser than the Atlantic, this one foot of water will consequently exert more pressure than the one foot of water of the Atlantic. We must therefore continue to pour more water into the Atlantic until its lateral pressure equals that of the Mediterranean. The two seas will then be in equilibrium, but the surface of the Atlantic will of course be at a higher level than the surface of the Mediterranean. The difference of level will be proportionate to the difference in density of the waters of the two seas. But here we come to the point of importance. In determining the difference of level between the two seas, or, which is the same thing, the difference of level between a column of the Atlantic and a column of the Mediterranean, we must take into consideration _only the water which lies above the level of the ridge_. If there be one foot of water above the ridge, then there is a difference of level proportionate to the difference of pressure between the one foot of water of the two seas. If there be 2 feet, 3 feet, or any number of feet of water above the level of the ridge, the difference of level is proportionate to the 2 feet, 3 feet, or whatever number of feet there may be of water above the ridge. If, for example, 13 should represent the density of the Mediterranean water and 12 the density of the Atlantic water, then if there were one foot of water in the Mediterranean above the level of the ridge, there would require to be one foot one inch of water in the Atlantic above the ridge in order that the two might be in equilibrium. The difference of level would therefore be one inch. If there were 2 feet of water, the difference of level would be 2 inches; if 3 feet, the difference would be 3 inches, and so on. And this would follow, no matter what the actual depth of the two basins might be; the water below the level of the ridge exercising no influence whatever on the level of the surface.

Taking Dr. Carpenter’s own data as to the density of the Mediterranean and Atlantic waters, what, then, is the difference of density? The submarine ridge comes to within 167 fathoms of the surface; say, in round numbers, to within 1,000 feet. What are the densities of the two basins down to the depth of 1,000 feet? According to Dr. Carpenter there is little, if any, difference. His own words on this point are these:—“A comparison of these results leaves no doubt that there is an excess of salinity in the water of the Mediterranean above that of the Atlantic; but that this excess _is_ slight in the surface-water, whilst somewhat greater in the deeper water” (§ 7). “Again, it was found by examining samples of water taken from the surface, from 100 fathoms, from 250 fathoms, and from 400 fathoms respectively, that whilst the _first two_ had the _characteristic temperature and density of Atlantic water_, the last two had the characteristics and density of Mediterranean water” (§ 13). Here, at least to the depth of 100 fathoms or 600 feet, there is little difference of density between the waters of the two basins. Consequently down to the depth of 600 feet, there is nothing to produce any sensible disturbance of equilibrium. If there be any sensible disturbance of equilibrium, it must be in consequence of difference of density which may exist between the depths of 600 feet and the surface of the ridge. We have nothing to do with any difference which may exist between the water of the Mediterranean and the Atlantic below the ridge; the water in the Mediterranean basin may be as heavy as mercury below 1,000 feet: but this can have no effect in disturbing equilibrium. The water to the depth of 600 feet being of the same density in both seas, the length of the two columns acting on each other is therefore reduced to 400 feet—that is, to that stratum of water lying at a depth of from 600 to the surface of the ridge 1,000 feet below the surface. But, to give the theory full justice, we shall take the Mediterranean stratum at the density of the deep water of the Mediterranean, which he found to be about 1·029, and the density of the Atlantic stratum at 1·026. The difference of density between the two columns is therefore ·003. Consequently, if the height of the Mediterranean column be 400 feet, it will be balanced by the Atlantic column of 401·2 feet; the difference of level between the Mediterranean and the Atlantic cannot therefore be more than 1·2 foot. The amount of work that can be performed by gravity in the case of the Gibraltar current is little more than one foot-pound per pound of water, an amount of energy evidently inadequate to produce the current.

It is true that in his last expedition Dr. Carpenter found the bottom-water on the ridge somewhat denser than Atlantic water at the same depth, the former being 1·0292 and the latter 1·0265; but it also proved to be denser than Mediterranean water at the same depth. He found, for example, that “the dense Mediterranean water lies about 100 fathoms nearer the surface over a 300-fathoms bottom, than it does where the bottom sinks to more than 500 fathoms” (§ 51). But any excess of density which might exist at the ridge could have no tendency whatever to make the Mediterranean column preponderate over the Atlantic column, any more than could a weight placed over the fulcrum of a balance have a tendency to make the one scale weigh down the other.

If the objection referred to be sound, it shows the mechanical impossibility of the theory. It proves that whether there be an under current or not, or whether the dense water lying in the deep trough of the Mediterranean be carried over the submarine ridge into the Atlantic or not, the explanation offered by Dr. Carpenter is one which cannot be admitted. It is incumbent on him to explain either (1) how the almost infinitesimal difference of density which exists between the Atlantic and Mediterranean columns down to the level of the ridge can produce the upper and under currents carrying the deep and dense water of the Mediterranean over the ridge, or (2) how all this can be done by means of the difference of density which exists below the level of the ridge.[75] What the true cause of the Gibraltar current really is will be considered in Chap. XIII.

_The Baltic Current._—The entrance to the Baltic Sea is in some places not over 50 or 60 feet deep. It follows, therefore, from what has already been proved in regard to the Gibraltar current, that the influence of gravity must be even still less in causing a current in the Baltic strait than in the Gibraltar strait.