202 



NATURE 



yjan, 2, 1879 



ZOPPRITZ ON OCEAN CURRENTS 



I SEND you a translation by a friend of an important 

 contribution to the theory of ocean currents by Prof. 

 Zoppritz, of Giessen, which has recently appeared in the 

 Annalen der Hydfographie und Marithnen Meteorologie. 

 The mathematical part of the subject has been published 

 in the Annalen der Physik for April last, a translation of 

 ■which will be found in the Philosophical Magazine for 

 September. 



One of the main objections urged against the theory 

 that ocean currents are due to the impulse of the winds 

 is that the winds can, it is alleged, produce only a surface 

 drift, whereas many of the currents extend to great 

 depths. I have always maintained that this objection is 

 totally erroneous ; that if the surface of the ocean be 

 impelled forward with a constant velocity by the wind or 

 by any other cause whatever, the layer immediately below 

 will be dragged along with a constant velocity somewhat 

 less. The layer underneath this second layer will in turn be 

 also dragged along with a velocity less than the one above 

 it. The same will take place in regard to each successive 

 layer, the velocity of each being somewhat less than the 

 one immediately above it, and greater than the one below 

 it. In this manner the surface velocity may be trans- 

 mitted downwards to any depth. This conclusion has 

 now been demonstrated by Prof. Zoppritz, in the follow- 

 ing paper, to be perfectly correct. James Croll 



Though for a long time the majority of seamen and 

 geographers have firmly held the opinion that the great 

 equatorial ocean currents derived their origin from the 

 trade winds, yet, so far as I know, no attempt has yet 

 been made to treat the physical problem of the propaga- 

 tion of surface-velocities downwards through a very thick 

 stratum of water, with the means presented by the theory 

 of the friction of fluids, as elaborated within the last 

 thirty years. Such an attempt is all the more demanded 

 as many authors have lately denied that surface-forces 

 could set the sea in motion to any considerable depth. 

 At the same time the most groundless assumptions have 

 been set forth as to the depth of such drift-currents. 



The essential principle of the theory of the internal 

 friction of fluids is that when a plane stratum of water is 

 moved forward, by any cause, in its own plane with a 

 given velocity, the adjoining stratum cannot remain at 

 rest, but, in consequence of its molecular cohesion ex- 

 periences an impulse to move in the same direction. And 

 if the velocity of the former stratum be continuous the 

 latter assumes a velocity which tends to approximate con- 

 stantly to the given velocity. This second stratum now 

 exerts the same ilnfluence on a third adjoining stratum 

 that it had to suffer from the first, and sets it in motion 

 in the same direction. The third stratum draws with it 

 in a similar manner a fourth, a fourth a fifth, and so on. 

 The propagation of the velocity is only bounded by the 

 limits of the fluid itself. If thess limits consist of a 

 -solid plane parallel to the strata, then the propagation of 

 the velocity will cease only at this point, i.e., between the 

 last liquid stratum and the first solid stratum. 



The law according to which two neighbouring strata of 

 velocities mutually influence one another has already 

 been demonstrated by Newton, and the accelerating 

 force exerted by the friction has been assumed as inde- 

 pendent of the pressure and proportional to the dif- 

 ference of velocity. The later theory of the friction of 

 fluids carries out this fundamental hypothesis as to the 

 propagation of velocity between strata of the same 

 medium which lie at an indefinitely small distance | from 

 one another, and have accordingly only an indefinitely 

 small difference of velocity A, inasmuch as it makes 

 the acceleration produced by the friction, at the plane in 

 which the strata meet, proportional to the quotient A : {. 

 The factor k, by which this quotient must be multiplied 



in order to give the acceleration, is called the Coefficient 

 of Interyial Friction. 



The Newtonian hypothesis can be applied likewise to 

 those parts of the bounding-surfaces of the fluid (where 

 it is in contact with other bodies) which may possess 

 independent motion. Here the acceleration produced by 

 the limiting medium (which may be solid, fluid, or 

 gaseous) is proportional to the difference of velocity 

 which may in this case be finite. The factor of the pro- 

 portion is called the Coefficient of Exte7-nal Friction. If 

 the bounding body is a solid or even a fluid, then the 

 fluid may wet it, that is, the stratum of fluid touching 

 the limiting body may cling so fast to that body as to 

 assume the same velocity. The coefficient of external 

 friction is in this case infinitely great. This is the case 

 between wood and water, glass and water ; and, on 

 the other hand, not so between glass and quicksilver. 



The theory founded on this simple hypothesis has been 

 subjected to the most varied experimental tests, and has, 

 on the whole, been found to agree with the facts, so that 

 the hypothesis may be regarded as proved. 



In order to apply this theory to ocean currents, the 

 simplifying presupposition has been made that the ocean 

 is a mass of fluid contained between two horizontal planes 

 at the distance h from one another, but in other respects 

 unbounded. On the surface of this mass of fluid a wind 

 of uniform strength and direction is acting at all times, 

 while the under-surface wets a solid plane, the sea- 

 bottom, and is therefore always at rest. 



We must not, however, look on the action of the 

 moving air on the surface stratum of the water as pro- 

 ceeding according to the Newtonian hypothesis; it will 

 act in this way only so long as the surface remains level. 



But the wind produces waves and acts on them accord- 

 ing to quite different laws. One fact of experience is 

 available here, viz., that the surface-stratum of the ocean 

 under the influence of a uniform wind, moves in the 

 direction of the wind with a constant velocity dependent 

 on the strength of the wind. If, therefore, we place on 

 the velocity of the water at the surface the condition that 

 it has a value w^ at all times given, everywhere uniform, 

 and of uniform direction, then the problem of the deter- 

 mination of the internal velocity becomes soluble. 



But the simplifying presuppositions here assumed are 

 almost realised in the central equatorial regions of the 

 great ocean ; the solution of the problem becomes, there- 

 fore, of deep interest. 



The following are the chief results of the solution : — 



If for an infinitely long time the surface-stratum has 



been kept at an unchanging velocity, then the whole 



mass of water is in a steady state of motion, i.e., a state 



which no longer varies according to the time. The 



velocity w is then dependent only on the depth x beneath 



the surface, and diminishes in proportion as the depth 



increases, till at the bottom it reaches zero. This relation 



is expressed by the formula 



h - X 

 ^ = «,,___. 



Naturally it is presupposed that no other causes, e.g., 

 displacing currents, affect the motion of the deeper 

 strata. If these deeper strata are kept by any foreign 

 cause whatsoever in steady motion in a direction exactly 

 opposite to the assumed motion, then at some point 

 between the highest and the deepest strata there lies a 

 plane where the velocity = 0. If this plane lies at the 

 depth /zj, then in the mass which lies above it the velocity 

 follows the formula 



//, - X 



w = Wo -^ , 



"1 

 and is therefore in the same condition as if the strata 

 that lie beneath were a solid mass. 



It is specially noteworthy that the velocity is inde- 

 pendent of the coefficient of friction, i.e., that in the 



