UC-NRLF $B Efi 2bfi p The Distribution of Ocean Temperatures along tlie West Coast of North America deduced from Ekman's Theory of the Upwelliiig of Cold Water from the Adjacent Ocean Depths. By Dr. Georgre h\ McEwen, Physicist of the Marine Biological Station of San Diegq. With 21 Texjt- figures. Sonderabdruck aus Internationale Revue der gesamten Hydrobiologie DDD und Hydrographie ddd Herausgeber: BJORN HELLAND- HANSEN (Bergen), W. A. HE RDM AN (Liverpool). 6. KARSTEN (H«lk), CHARLES A. KOFOID (BerKc ley), L. MANQIN (Paris) SIR JOHN MURRAY (Edinburgh), ALBRECHT PENCK (Berlin), E. M, WEDDERBURN (Edinburgh). C. WE S E N- B E R Q - LU N D (Hillerod), FRIEDR. ZSCHOKKE (Basel) und R. W0LTERECK(Leipzig.Qaut2sch), RedaRteur Verlag von br. Werner KlinK^ardt, Leipzig 1912 f^MT The Distribution of Ocean Temperatures along tlie West Coast of Nortli America deduced from Ekman's Tlieory of tlie Upwelling of Cold Water from tlie Adjacent Ocean Deptlis. By Dr. Greorge F. McEweii, Physicist of the Marine Biological Station of San Diego. With 21 Text-figures. Introduction. The presence along the west coast of North America of a belt of cold surface water having at any point a much lower temperature than is normal for the corresponding latitude, has long been known. And several papers have been written in which a diversity of merely qua- litative explanations of this interesting and perplexing phenomenon have been given. The present paper is an attempt to explain quantitatively the temperature distribution by means of a new theory of oceanic cir- culation, developed by V. W. Ekman(l) of Kristiana. The contents of this paper fall under the following nine heads: I. A brief summary of some important and generally accepted facts concerning oceanic temperatures and circulation. II. A brief review of the theories that have been proposed to account for the cold-water belt along the west coast of North America. III. An abstract of the most important part of Ekman's theory of oceanic circulation needed in attacking the above mentioned problems. IV. Some general qualitative applications of his theory to a variety of temperature problems. V. The formulation of a temperature problem in such a way that a quantitative estimate of the mean monthly surface-water temperature for any given place can be made by means of the physical theory of heat and circulation. 274298 244 ' ' ■ ' " * G; p. McEwen. VI. TM soiiilrionof tM^iboVe "problem for four very different regions along the Pacific Coast, and a comparison of the observed and cal- culated values. VII. A discussion of the results, and additional test of the theory using the observations made by the Marine Biological Association of San Diego in a much more limited area. VIII. Some remarks on the influence of ocean temperatures on the coast climate of California. IX. Summary and conclusion. I. Some generally accepted facts regarding oceanic temperatures and circulation. Under normal conditions, such as prevail in mid-ocean, the surface temperatures are subject to periodic changes with the seasons similar to the temperature^) variations in the air over the land as shown by the curves (Figs. 1 and 2). But the range of the ocean temperatures is only about half as much. The estimated daily range for the ocean is about 2^, while that for the air is about 10**. Observations show that in the ocean the daily temperature change does not extend below a depth of 10 to 20 meters, and the annual change can not be detected below 200 meters. The curve (Fig. 3) illustrates in a general way how the temperature varies with the distance below the surface. Water more than 200 meters below the surface has a temperature decreasing from 10^ as the distance downward increases, often reaching the value zero on the ocean bottom, even at the equator. But the presence of several limited regions in which the temperature distribution differs notably from the normal, both v\dth reference to depth and time of year has long been recognized, and considerable discussion relative to the reason for their existence has been carried on. The question of the distribution of temperatures in the sea is so intimately connected with that of the character of its currents that it is practically impossible to separate them entirely. Consequently, as soon as one leaves the general question of the mean temperature of a given area, and wishes to decide whether that temperature is normal, and if not, from whence and how it is derived, the features of oceanic circulation must at once make part of the discussion. All temperatures are in the Centigrade scale. Ocean Temperatures along the West Coast ol North America. 245 As the motion of oceanic waters is partly determined by their tem- peratures, so their paths may often be traced out by the isothermal curves showing the surface temperatures. As the motion, when normal, is usually less accurately measurable than the temperature, and is much :iiO / ^ / \ \ \ / [ 1 \ \ 1 \ \ """"• N, \ / / \^ \ / 1 \ / V / \ / / — f \ ■ -- \ / / \, V \ \ I 9 10 Montlis I y -> s \, 1 1 \ I \ \ u \ 'I \ ,\ \\ / i \ , A K / / \ \ \ / \ "-/ \ \ 1 \ / \ / v '^>-^ 9 10 II 12 I Fig. 1. Fig. 2. Curve for Surface Temperature of the Curve for Surface Temperature of the Ocean, Lat. 32«30', Long. U0\ Ocean, Lat. 37«45', Long. 140«. Curve for Air Temperature at Yuma, Curve for Air Temperature at Sacra- Arizona, where the Latitude is the same. mento, where the Latitude is the same. more rapidly lost, it often happens that the distribution of current water can be more accurately determined by a study of its temperature than in any other way. Thus, by means of its temperature, the existence of a given current can be determined with certainty over an area far exceeding that in which it can be proved to have a perceptible constant motion in any given direction. 246 G. F. McEwen. Conversely, if a large body of water be shown to have a nearly uniform summer temperature, corresponding in general with the normal value for the latitude, and with local circumstances in its particular portions, this is, of itself, evidence that no large body of water intrudes within its borders from a region of a different normal temperature. In other words, in the general oceanic circulation, a stream of water with a temperature normal to one latitude can not move to a region where another temperature is normal without exhibiting its presence by a de- flection of isothermal curves. 20 14 12 foio r Sun tmt r 1 \ Wir ^ten v i-^. •■^iiiou. \ t \ s s "^ ""^ -~— < H -~». „,_ 0 200 400 600 800 ICOO 1200 1400 1600 1800 2000 100 300 500 700 900 1100 1300 1500 1700 1900 Depth in Meters Fig. 3. The source of the energy required to set and keep in motion the vast mass of ocean water has been productive of endless discussion. The attractive force of the moon, the lag of the water itself, the difference in temperature and density of the equatorial and polar regions, the unequal distribution of the atmospheric pressure has each in its turn been proposed and strenuously advocated as the true and only cause of ocean currents. To the seaman, however, the cause of the ocean currents has always been the winds, the motion of the waters of the sea taking its origin in the region where the winds attain their maximum constancy, that is, in the region of the trade-winds. The following results of Zoppritz's (2) theory of ocean currents have been widely used by hydrographers. A perfectly steady wind acting continuously on the surface of the sea will thru friction, give rise to a movement of the surface waters in the same direction as the wind, itself. If the latter continues for a s.ufficient length of time, the Ocean Temperatures along the West Coast of North America. 247 impulse, first felt only at the surface, will gradually communicate itself downwards, owing to the viscosity (internal friction) of the water, and the lower strata to a successively greater and greater depth will thus partake of the movement until it is finally shared by the whole mass, the velocity diminishing as the depth increases. The rate, however, at which the motion is communicated to the depths of the ocean is ex- ceedingly slow. For example, Zoppritz (2) estimated that in a depth of 4000 meters a surface current of a given velocity would require a period of about 100000 years to transmit a velocity of one half this amount to a point half way toward the bottom, and 240 years would be needed for the layer at the depth of 100 meters to attain one half the surface velocity. A period of 24 hours is required before a change in the direction of the wind affects the water at a depth of 5 meters. Similarly, when once established, these submarine currents exhibit a corresponding reluctance to undergo any change in direction or intensity. In general, according to Zoppritz, the time (t) in years required for a given constant surface velocity to produce a velocity one half as great at a depth of (x) meters is given approximately by the formula t = 0,024 X-. But observations show that perfectly steady winds do not exist, even in the region of the trades. The winds are constantly changing, and the surface currents change with them. The lower strata of the ocean, however are insensible to these changes, and at a considerable distance below the surface, the waters of the ocean have probably a slow but steady motion, the direction of the motion probably agreeing closely with the resultant surface winds. James Page (3) summarises the conclusions based on Zoppritz's theory as follows. We have, therefore, in the body of the sea two distinct sets of currents; first, those at the immediate surface, which move practically at the obedience of the surface winds, sometimes in one direction and sometimes in another, second those of the lower strata, which are constant in direction and velocity and represent the aggregate effect of the winds that have blown for ages past. II. Review of the theories that have been proposed to account for the cold-water belt along the west coast of North America. 1. Dall (4) has discussed the dependence of the cold in-shore water on the Japan Current somewhat as follows. The Japan Current is pro- duced by the impinging of the Pacific North Equatorial Current on the 248 Cr. F. McEwon. eastern shores of Formosa and the adjacent islands. While the larger part of the Equatorial Current passes into the China Sea, a portion of it is deflected northward along the eastern coast of Formosa, until reaching the parallel of 26^ it bears off to the northeast, washing the whole soulheastern coast of Japan, and increasing in strength as it advances to a limit which appears to be variable. At the very outset the Japan Current must force its way thru the barrier of the Loochoo Islands and little later thru that chain of rocks, shoals and islets extending from Yokohama to the Bonin Islands, then it has nearly 6000 miles to traverse before reaching the opposite shore of the Pacific. Thruout this distance it is opposed by the northeast monsoon from the end of September to the end of February. Hence it is not surprising that its force should be checked, and its continuity as an eastward current should be for the time almost obliterated. In fact, according to many reports of navigators the current is subject to serious fluctuations which appear to be due to the monsoons, and as compared to the Gulf Stream is cooler and has a much smaller volume. After reaching the northwest coast of North America near Sitka, Alaska it has a southward branch w^hich, by the time it reaches San Francisco has become a cold current rather than a warm one, simply because it is intruding on a normally warmer area. 2. Richter (5) attributed the low in-shore temperatures to a polar current, and his argument may be summarised as follows. The question not as to the existence, but as to to the character of the ocean cur- rents contiguous to the coast of California is still an open one. Some of the most recently published maps show that a cold current of great width washes our shores, and others again indicate that it is the de- flected warm Japan Current which is passing this country in its south- ward movement. A third opinion gives the surface w^aters to the "Kuro Siwo" or Japan Current, and identifies the sub -stream with the polar current. But his own conclusions are briefly as follows: The northern or arctic currents are powerful enough to alter materially the direction of the Japan Current. They sweep against the warm water as the polar waters meet the Gulf Stream on the north of Scotland. The arctic waters predominate on the surface by superior force until the "Kuro Siwo" gives a strong wall, which causes the cold current to pass underneath in the direction of the equator. One branch of this artic current continues south along the coast as far as Point Conception, latitude 34^. Ocean Temperatures along the West Coast of North America. 249 The bulk of the Kuro Siwo trends eastward, bat perhaps nowhere washes the shores of the United States, being separated from them by the narrow cold stream, and yet is near enough to exercise a powerful influence on the coast climate. Both streams flow to the south and produce a narrow eddy current next to the shore flowing north. The directions of the observed currents can, according to Richter, be accounted for on the basis of Zoppritz's theory from the fact that the coast of the United States tends northeastward^) from Cape Men- docino, latitude 40^ to Tatoosh Island, latitude 49^, and tends south- eastward to San Diego, latitude 32^ 45', while the prevailing winds are easterly near Tatoosh Island and westerly near San Diego. 3. Bishop (6) assumed a flow of cold antarctic water northward along the ocean bottom, and explained how it would cool the in- shore water as follows. It has been the custom to call the cold stream which is of a very low temperature, immense volume, and great velocity, and flows along the west coast of North America, a con- tinuation of the Japan Current, but there are two insuperable objec- tions to that solution. First, the Japan Current must necessarily pass out and dissipate itself in the vast breadth of the Pacific Ocean long before reaching Alaska. Second, the Japan Current is entirely too warm, and the Alaskan glaciers are inadequate to materially cool the adjacent waters. But the antarctic continental glacier extending 4000 miles in length along the antarctic circle cools the ocean water, and because of its increased density this cold water sinks to the bottom and flows north- ward, being forced on by the accumulated supply behind it. After passing the Tropic of Cancer, the Pacific Ocean contracts in breadth. In latitude 45^, it is only half as wide as at the equator. Hence, the northward moving water is accelerated to twice its previous velocity. From the rotation of the earth this current would be forced to the east against the North American continent in the latitude of Sitka and Vancouver's Island. The resulting pressure drives -the water up the continental slope and it then flows southward, having no other outlet. It continues along the coast, being held there in opposition to the west- ward deflecting force due to the earth's rotation by the eastward pressure due to the bottom current which continually pushes up from the deep sea under the easterly globe-thrust along the entire west coast of North America. ^) The coast actually trends slightly to the west. Revue d. ges.Hydrobiol. u.Hydrogr. Bd. V. H. 2/3. 17 250 Gr. F. McEwen. 4. Dall (7) refuted Bishop's argument and reasoned as follows in favor of the commonly accepted view (8), that the Japan Current is a cold stream bordering the California coast and has a relatively low tem- perature because of its passage thru high latitudes. The northeast trades, blowing hard and steadily for ten months in the year carry the warm water, w^hich the Japan Current delivers in mid-ocean, to the northwest coast of North America which it reaches at latitude 54^ near Sitka, Alaska. There, the stream divides into two branches, a northerly and a southerly one. At this point the maximum temperature is 20*^, but the average is about 15,6 ^ As it moves down the coast, it loses its heat, and produces the fogs and rains of the Oregonian regions cooling off so that when it reaches the latitude of the Golden Gate, it is colder than the sea water under normal conditions in that latitude. That this essentially superficial stream is not due directly to the impinging of cold antarctic water on the northwest coast seems to be certain from the fact that the temperature of the latter is nearly zero, while the current when it reaches the coast is 17^ or more warmer than that; and that the water of the current is warmer in latitude 54^ than it is in the more southern part of its course, whereas, if it was abyssal, we should expect it to be colder, and to gradually warm up as it moved south-ward exposed to the action of the sun. The Pacific is open without stint to the influx of antarctic cold water, and in it probably goes on a great system of true oceanic cir- culation such as might occur were there no continents. The general system of oceanic circulation is influenced by the rotation of the earth, differences of density and temperature in the oceanic mass, tides, the pressure of the atmosphere, and various minor causes, and there are no indications that would lead us to believe that such movement are other than very slow and gradual, or that they have any marked effect in producing the superficial streams of rapidly moving water which we call currents. 5. Hoi way (9) showed that the hypothesis of a surface stream flowing south and being abnormally cold because of its passage thru high latitudes can not be reconciled with observations, and attributed the cold surface water to an upwelling of bottom water as follows. The cold in-shore water cannot be due to a surface current flowing parallel to the coast, as observations of the surface temperatures have indicated narrow belts of alternately warm and cold water directed normal ly to the coast, the temperature differences being 2^ or 3*^. Also the coldest part of the cold-water belt is near Cape Mendocino latitude Ocean Temperatures along the West Coast of North America. 251 40^, there being warmer water both to the north and to the south of this region. But a cold current flowing down from Alaska would continually rise in temperature as it moved southward. The only re- maining explanation is that there is a belt of cold water upwelling from the adjacent ocean depths. The above surface temperature relations present two problems. First, what is the cause of this cold-water belt? We have already seen that it must be due to an upwelling from the adjacent ocean depths, but what is the cause of this upwelling? And secondly, why should the coldest portion of this area be in the vicinity of Cape Men- docino, instead of farther north? An drees (10) attributed the cold-water areas to a vertical current caused by the winds blowing off shore and driving the surface water to the leeward, thus causing a return drift at the bottom of the ocean and an upwelling near shore. Hann (11) accepted this view and added ''the sharp deflection of an ocecan current off shore may cause a rise of cold water from below". He also assumed that the whole coast of California belonged to the area of the constant trade winds. Buchanan also accepted the theory of An drees. But observations show that the trade-wind region does not extend as far north as California and the component of the wind velocity normal to the coast blows toward the land except for a few weeks in the winter time, the season when the temperature is nearest to the normal. Another peculiar fact concerning the temperature distribution is shown by the charts made by Sir John Murray (12). They indicate that water temperatures over the continental shelf are notably less than those at the same level but farther out over the great ocean depths. The observations and hypotheses referring to the relatively cold water lying along the west coast of North America may, according to Holway, be summarised as follows: a)^) The definite belt of cold water can not be traced south of Point Conception. b) In the summer, the coldest part of this belt is in the vincinity of Cape Mendocino. c) The source of this cold coast water is in the ocean depths to the northwest of Cape Mendocino. ') The observations a) and b) and the theory of Zoppritz have an important bearing on the hypotheses c) and d). 252 Gr. F. McEwen. d) This cold water at or near the ocean bottom has a slow drift agreeing in direction with the average direction of the surface drift, and is driven to the surface on striking the slope of the continental shelf. Local variations in the temperature of the cold coast water are due to the submarine valleys and other irregularities in the slope of the con- tinental shelf. 6. A study of the preceeding investigations shows that the general method employed was to first examine the data and then to formu- late a hypothesis to account for the observed results, the hypothesis being supported partly by the fact that it accounted in a qualitative way for the particular observations used, and partly by the theory of Zoppritz. Some writers used the effect of the earth's rotation and others did not. Sometimes erroneous statements regarding the observed facts were used in support of the explanation. The temperature distri- bution was regarded as constant thruout the year. No quantitative tests were attempted, and the explanations given by the different writers are inconsistent with each other. So the question is still an open one. III. Ekman's theory of oceanic circulation. It is well known that the motion of large masses of water in the ocean is accompanied by irregular vortex motions that cause the com- putation of the actual frictional forces according to the methods of rational hydrodynamics to be worthless. There is, in fact, an obvious disagreement between the results of hydrodynamics on the one hand and experience on the other. Therefore, for practical purposes we have had to be satisfied with purely empirical formulae, which because of their very limited field of application, have been unable to afford any help to oceanography. But all of the factors affecting the motion of ocean water can be taken into account if problems of a sufficiently simple type are formulated. By devising a series of such simple typical problems and solving them by exact analytical methods in which all of the factors are used, a sort of framework can be formed about which a theory of ocean currents corresponding to the different actual problems can be built. The actual problems, however complex they may be, can then be most readily attacked, not by analytical methods, but by suitably combining the proper typical problems that have been solved. By proceeding in this way a number of very important results have been obtained which are the immediate consequences of the general Ocean Temperatures along the West Coast of North America. 253 principles of mechanics, and are therefore independent of any hypothesis as to the laws of fluid friction. The following discussion will include only a brief statement of the assumptions made by Ekmann (1), and a few of the most important results that he obtained. It is a well known fact that the rotation of the earth upon its axis tends to deflect (13) a body moving along its surface to the right of its path in the northern hemisphere and to the left in the southern hemisphere. Therefore, in addition to the forces given by the conditions of the problem, this deflecting force must be introduced. It can be proved that a correct solution of a problem con- cerning the motion of a body on the earth's surface will result if we assume no rotation of the earth upon its axis, but introduce the force. 1. F = 2Va)sin(p into the equations of motion and solve in the usual way. Velocities and displacements are measured relative to the earth's surface, (V) is the velocity of the body (w) is the angular velocity of the earth, (0) is the latitude of the place, and (F) is the force perpendular to the direction of motion, as shown in (Fig. 4). F Fig. 4. That is, the deflecting force due to the earth's rotation is pro- portional to the velocity of the body relative to the earth's surface and increases from zero at the equator to a maximum at the poles, and for a horizontal velocity in any direction is directed normally to the motion. (Fig. 4) corresponds to the northern hemisphere, but the arrow (F) would have to be reversed to correspond to the southern hemisphere. First typical problem. Imagine a large ocean of uniform depth and without differences of density affecting the motion of the water. The influence of neighboring ocean currents and continents is left out of account so that water can freely flow in or out of the region con- sidered. Suppose the water surface to be impelled by a steady and uniform wind equal in strength and in direction over the whole region. 254 G. F. McEwen. and that these conditioDs have continued long enough to estabHsh a stationary state of motion. Introducing the deflecting force (F) into the equations of motion of a viscous fluid and solving in the usual way gives the following results for the northern hemisphere. The velocity of the surface water is directed at an angle of 45^ to the right of the Fig. 5. The arrows show the velocities of the water at the depths 0,0 D, 0,1 D, etc., below the surface. Vo is the surface velocity. wind velocity (Fig. 5). The magnitude of the velocity of the water decreases as the distance below the surface increases, the direction of the velocity of each layer being to the right of that above it. At the depth 2, D = TT \f — L \ q CO sm * where {[i) is the coefficient of viscosity and (q) equals the density, the magnitude of the velocity is only ^/oo of that at the surface, and is in the opposite direction. This distance (D) is called the "depth of the wind-current", and the motion at greater depths is assumed to be negligible. Another result is 3. V = '^- — ^ |/2,aqw sin^ where (Vo) is the surface velocity of the water and (Tj) is the tangen- tial pressure at the surface due to the wind. The general character of the motion may be illustrated as follows. Imagine a spiral stairway so situated that the edge of the top step is Ocean Temperatures along the West Coast of North America. 055 directed at an angle of 45^ to the right of the wind velocity, thus coinciding with the arrow (Vo) of (Fig. 5). Now, if as we descend, the successive edges are shortened so as to have the successive lengths of the arrows of the diagram (Fig. 5), each edge will represent in magnitude and direction the velocity of the water at that depth. And by the time a half turn had been made the edge of the step, that is, the velocity of the water would be only ^/ao of its value (Vq) at the top, and from there downward the velocity would be still smaller. So, for practical purposes we can neglect the motion below that point. The total momentum of the "wind-currentu is directed at right angles to the wind itself. The flow or volume of water per unit time transported^) parallel to the wind is zero, and that perpendicular to the wind is 4^ VoD _ T, nY'i 2qft) sin^* Therefore, the direction and magnitude of the flow depends only on (Ti) and not upon (p). Within this surface layer of thickness (D) it is evident from (Fig. 5) that water is actually transported in various directions and that the amount carried per unit time in a layer of given thickness decreases as the depth increases. But when we add up the amounts flowing parallel to the wind in the whole layer, it is found that there is as much water flowing with the wind as against it, so the total quantity per unit time, that is, the flow parallel to the w^ind is zero. While when we sum up the amounts moving normal to the [V D 2 1 [—Tf^) = -Q (Vo D)J is obtained, the direction or the flow being to the right of the wind. That is, a uniform wind blowing over deep water in a region remote from any obstructions, will trans- port water only at right angles to its own direction, the flow being the same as if the velocity of all the water in the layer of thickness (D) was directed to the right of the wind and the magnitude was 7o of its value at the surface. The experimental value of (fi) for laminar motion is about 0,014, and when substituted in (2) gives the value 44 5. D = ^/-^— :^ centimeters. y sm* The theory of Zoppritz is based on exactly the same assumptions as the above except that the influence of the earth's rotation is neglected, *) Thru a vertical rectangle of unit width and perpendicular to the direction of the motion. 256 G. F. McEwen. and his result is entirely different. But as was stated before, the motion is not laminar, but is turbulent and the ordinary value of (f^) which gave the absurdly small value of (D) must be replaced by a virtual value much greater than 0,014. The virtual value 6. ^ ^ '-—, would be different under different conditions of wind, velocity etc., and can only be determined by current measurements and other observations carried out under varying circumstances. From the rough measurements now available^), a mean value of (D) would be 75 meters. Experiments on the relative velocities of the wind and surface current led to the following approximate relation: 0,0127 ^^1 y sin * where (V^) is the wind velocity and (Vq) that of the surface water. That is, at the latitude 45^ the velocity (Vq) of the surface water is approximately ^/ss times that of the wind (V^y) which causes it. This multiplier would be about 0,013 at the poles, and would increase as the equator is approached. If the wind velocity is in miles per hour and the value of (Vo) is required in meters per second the formula becomes 0,00569 ^^ 8. Vo = 7^=F== Vw. y sm * From Ekmans theory, the time required for a steady current to produce any fraction of the final limiting value is independent of the value of (fi) and the current would be practically fully developed in 24 "pendulum hours''^). And thus outside of the tropics where this theory does not hold, only a few hours are required to set up a stationary state of motion, and the enormous times that Zoppritz computed on the basis of laminar motion and the ordinary value of (fi) are meaningless. Second typical problem. Assume as before an infinite ocean of uniform depth (d), [(d) is greater than (D)] and of uniform density (q) Suppose the surface to be inclined at a constant angle (cP). Solving as before, the following results were obtained. The current will consist ^) Ekman deduced the following approximate formula for deter- mining (D). Ocean Temperatures along the West Coast of North America. 257 of a "bottom-current" of thickness (D) running more or less in the direction of the force, and above this a current reaching right up to the surface with the almost uniform velocity Q TT = _A^HL^_ y- ^^ 2tosin* perpendicular to the force, w^here (g) is the acceleration of gravity, and (0) is the inclination angle of the surface with a horizontal. In order to represent the result graphically, it is sufficient to so turn (Fig. 5) that the longest arrow points perpendicularly to the left of the pressure gradient. Then add to each arrow in turn the constant velocity (Uq) to the right. The successive resultants will be the velocities at the distances 0,0 D, 0,1 D, 0,2 D etc. above the sea bottom. (Fig. 6) has been constructed in this way, (OY) being the direction of the pressure gradient. In the bottom -current the flow in the direction of the pressure gradient is 10. Sy = ^ = 0,159 UoD, while the flow normal to the pressure gradient is 11. Sx = ^^^ UoD = 0,84 UoD. The significance of the above problem may be further brot out by the following explanation. Fluid motion can also be generated by difference in pressure, and in a region of the ocean far removed from obstructions and having a uniform depth greater than (D), it follows from Ekman's theory that if the pressure decreases as shown by the arrow in (Fig. 6) the resulting motion from the upper surface downward to a distance (D) above the bottom will be uniform, tho not in the direction in which the pressure decreases but at right angles to that as shown by the arrow (Uq) of (Fig. 6). And the motion between the bottom and a surface at the distance (D) above the bottom can be represented by another spiral stairway, the edge of the top step coin- ciding with (Uo) and the successive edges decreasing to zero at the bottom by having the succession of values of the arrows in (Fig. 6). By summing up, as before, the amounts flowing parallel to the pressure gradient and then the amounts normal to the gradient in the different layers of this "bottom stream" it will be found that the total flow or volume per unit time transported in the direction of the gradient is 0,159 UoD while the flow in a perpendicular direction is 0,84 UqD. -s ^ J 1 u 1 sideral hour *J 1 pendulum hour = : — -=■ . sm* 258 G. F. McEwen. Third typical problem. Assume a steady uniform wind blowing in a constant direction everywhere outside a straight and infinitely long coast. Under these conditions, the depth of the ocean being supposed uniform, the current would be the same at any two places at the same distance from the coast, and no inclination of the surface can occur in the direction of the coast itself. Perpendicular to this direction a slope will arise and gradually increase until the total flow normal to the coast is zero. If the depth exceeds (2 D) there will be three distinct currents: first, a "bottom-current" of depth (D) moving more or less in the direction of the slope, but with a deflection to the right increasing from 45^ at the hot tomto 90^ at the top: second, a "mid- water-current" of almost uniform velocity parallel to the coast and reaching from the top of the "bottom-current" to the depth (D) below the surface. (The velocity of this current is proportional to the com- ponent of the wind velocity parallel to the coast): third, a "surface- current" in which the velocities are equel to those of a wind- current" superposed on the velocity of the "midwater- current" „The bottom - current" and the "surface-current" will not be appreciably in- fluenced by an alteration of the depth (d) as long as it exceeds (2D), and the only effect then will be a corresponding alteration of the depth of the uniform "mid water-current". The most striking result of the coast's influence is that a wind is able indirectly to produce a current more or less in its own direction from the surface down to the bottom, while in the absence of coasts Ocean Temperatures along the West Coast of North America. 259 the wind's effect would be limited to a comparatively thin surface layer, even if blowing steadily for any length of time. The general circulation of the water in the neighborhood of a coast is shown in (Fig. 7) which represents a cross-section perpendicu- lar to the coast. The wind is assumed to blow parallel to the coast, and normal to the paper from the reader. The arrows show the com- ponents of the velocity of the water in a plane perpendicular to the coast. The actual velocity would be the resultant of that shown and a component parallel to the wind. Fig. 7. Cross Section Normal to the Coast. If the wind should blow in the opposite direction to that shown, the motion of the water would be reversed. Thus the component of the wind velocity parallel to the coast causes an upwelling of bottom water under the conditions shown, or it carries surface water to the bottom if its direction is reversed. The rate at which water is carried up or down is proportional to the magnitude of the component of the wind velocity parallel to the coast ^). From a computation of the time required for the stationary motion to be established, the following general conclusions may be drawn. Assuming (D) = 75 meters, the stationary state of motion will be prac- tically fully established to within several hundred kilometers from the ^) As the depth (d) diminishes from the value of (D) in problems 1 and 2 or diminishes from (2 D) in problem 3 the motion becomes less influenced by the earth's rotation, that is the flow due to a wmd tends to have the same direction as the wind, and that due to a pressure gradient tends to flow from a region of high to one of low pressure. But no important alteration in the results worked out above will follow until (d) is less than (0,5 D) in 1 and 2, or less than (D) in 3. 260 Gr. F. McEwen. coast in a few days if the depth does not exceed about 400 meters (that is over the continental shelves). In the deep ocean, on the other hand, particularly in the case of very broad currents, say 1000 or more kilometers in width, the "mid water-current" may require several months to become approximately fully developed^). Thus the effect of a wind is entirely different from what Zoppritz's theory shows, and the time required for producing the effect is measured in days or months, rather than in geological periods. IV. Qualitative applications of Ekman's theory to several peculiar temperature distributions. Since, if the preceeding theory is true, it is possible for a wind acting under proper conditions to bring bottom water to the surface, and since we know from observations that the bottom water is cold, abnormally low surface temperatures may be due, at least in some cases, to a combination of circumstances which would produce an up- welling of the cold deep water into the warmer surface water. H. Thorade (14) has compiled, from all reliable sources available, a large amount of detailed information concerning the prevailing wind directions and surface temperatures over the North Pacific Ocean cor- responding to each month of the year (Figs. 20 and 21 are copies of two of hi^ temperature charts, and they also indicate the prevailing surface currents for the months of Jan. and Aug. respectively). He assumes that the low temperatures near the coast are due to the up- welling of cold bottom water, and points out a close correspondence between the prevailing wind directions and the temperature distribu- tion He does not give the velocity of the winds, but only the pre- vailing directions, and devotes only one page to a qualitative applica- tion of Ekman's theory, the results of which are shown to agree in a general way with the observations on the seasonal distribution of temperatures. This is the first application of the new theory of up- welling to the California region. The statement made by Hoi way (9) that during the summer the ^) The approximate value of the time required for such a current to attain 0,7 of its final value is given by the formula where (x) is the distance from the coast in kilometers across the stream and (d) and (D) are depths in meters. Ocean Temperatures along the West Coast of North America. 261 coldest surface water was in the vicinity of Cape Mendocino, lat. 40^, is fully verified by the maps in Thorade's (14) article. And from the present theory we would expect the water there to be colder than that farther south because of the higher latitude and the upwelling combi- ned. North of this region the wind component parallel to the coast is directed to the south but rapidly diminishes, being only 2 or 3 miles per hour off Vancouver, less than 500 miles farther north, while the velocity at latitude 40^ is about 15 miles per hour. So we would expect the quantity of cold water upwelling to diminish proportiona- tely in that distance. The normal temperature for the latitude of Vancouver is about 2^ more than the actual temperature off Cape Mendocino from June to September, but is about 2^ less during the rest of the year. The charts show as would be expected that off Vancouver the temperature is practically independent of the distance from the coast. The presence of abnormally cold water on the continental shelf, referred to in Hoi way's article, would result directly from Ekman's theory, as the prevaihng wind direction is such as to drive the bottom water toward the coast over this slope. Along the west coasts of Africa and South America the prevailing wind directions (15) are such that a temperature distribution similar to that off the west coast of North America would be expected on the basis of Ekman's theory, remembering that in the southern hemisphere the deflecting force is directed to the left of the motion. This agrees with the observations which indicate a reduction of in- shore tempera- tures corresponding to the strength of the wind component parallel to the coasts. The qualitative agreement being so satisfactory, it seemed very desirabler that a detailed and quantitative test should be made. And I have attempted to so formulate the problem of the distribution of ocean temperatures along the west coast of North America that it could be put into mathematical language and solved with the aid of Ek- man's equations of fluid motions. V. The mathematical formulation of the temperature problem for a given locality. Assumptions. The normal in-shore temperature for any latitude is the same as the actual temperature at a point in mid-ocean having the same latitude. 262 Gc. F. McEwen. The difference between the actual temperature and the normal temperature is due entirely to the mixture of cold water from the ad- jacent ocean bottom with the surface water. The cold water up welling in a particular region is due entirely to the winds in that latitude, and no cold water from the surface in other localities enters the region. Observations and deductions. An examination of the iso- therms of Thorade's charts showed that the temperature of a surface layer of water is approximately constant for a certain distance (xi) out from the coast, and increases in proportion to the distance from that point out to a point whose distance is (X2) from the first, and re- mains nearly constant from this point to the limit of his map, longi- tude 140 ^ Therefore the amount of heat in this layer of unit width and thickness (y) would be y{xiT + ix2(T + t.) + X3tJ where (T) is the actual in-shore temperature, and (t.2) is the normal temperature for the latitude. Assume that this amount of heat is the same as if the total volume y {xi -{- x^ -f- X3} was, to begin with at the normal temperature (to) and a volume (xy) at the temperature (to) was then removed from the region and replaced by an equal volume (xy) up welling from below at the temperature (tj. (T) and (to) denote mean monthly temperatures corresponding to the latitude, and(xy) will be assumed to be the volume upwelUing into the surface layer during the month's time ending with the date to which (T) and (to) correspond. Equating the two expressions for the amount of heat gives the equation 12. y{xiT+yX2(T + t,) + X3to} = |(xi + X2 + X3 — x)to + xti}y. The actual temperature at any instant is the result of the con- tinuous action of various causes, absorb tion of heat from the sun, radiation, evaporation and the intrusion of cold water. But, in order to simplify the computation, it is assumed that the actual continuous process can be replaced by the following artificial one. Assume that whatever the temperature for any given month may be; the temper- ature a month later would be normal were there no intrusion of cold water; and assume that if all the water that actually intruded during the previous month were then quickly mixed with the surface layer the resulting temperature would be the same as the actual temperature. A time interval of one month was adopted because the computation of the mean temperature for each month can then be made directly, and Ocean Temperatures along the West Const of North America. 263 compared with the observed temperatures, which are given for the same time interval. From equation (12) the following equation for calculating (t), the difference between the normal temperature (to) and the actual tem- perature (T) was obtained: where (x) is the length, normal to the coast that multiplied by the cross section (yXl) of the volume considered, gives the amount of cold water intruding, and (X1 + X2) is the distance from the coast out to the point where the temperature is practically normal. Now if this quantity (x), which depends upon the amount of water upwelling in one month, can be determined, then the temperature reduction (t) can be calculated by substituting in equation (13) if the normal temperature and that of the upwelling water are known. With the aid of Ekmans theory (x) can be computed as follows. The flow normal to the coast due to the surface current between the depths (zi) and (zo) is 14. s= / udz where (u) is the velocity normal to the coast. From Ekman's equation this reduces to /5Z2 15. s = I Vo e"^^ cos (45^ — az) dz *^Zl where ^/"qa> sin^ n 1^- ^=V — jr—=-D Integrating between these limits and simplifying the result we have 17. S = ^ (1 — 2-k^ cos kn) = approximately, ^) ^- (k^r - !^) where a surface layer down to the depth (kD) is considered. Observations show that the temperature is practically uniform from the surface down to the depth of about 5 meters, so the value of (y) can be taken equal to 5 meters or 0.06 (D) where (D) is 75 meters. Substituting this value of (k) in equation (17) gives 18. S = ^'^fJ' ^ = 0,0418 Vo D, and V271 ' ^) This approximation is true only for small values of (k7r), the error being 264 G. F. McEwen. S V 19. xi = - = 3.1 Substituting for (Vq) its value in terms of the wind velocity from equation (8) we have 90 1^ 3,1 (,0057) ^, ^ 0,004 7r|/2V'sin^ '^ /^i^ ^ where (x^) is the average velocity in meters per second at which a surface layer of thickness 5 meters leaves the coast and (V^) is the component of the wind velocity parallel to the coast in miles per hour. Multiplying by the number of seconds in a month we have 21. . - = ^V.. y sin* Substitute this value of (x) in equation (13) and the result is 22. t= MI^^(t,-t,) Vsin* (xi+yXaj which is a theoretical relation between (t) the reduction of the tem- perature below the normal value, the wind velocity (V^) parallel to the coast, the normal temperature (to), and the temperature (ti) of the upwelling water which causes the abnormally low actual temperature (T). For any given place the latitude ((^) is constant, and observation shows (ti), (xi), and (xo) to be practically constant. So the only va- riables for a given locality are the wind velocity parallel to the coast and the normal temperature. The wind velocity must be in miles per hour and the distances (xi) and (xo) in meters. VI. The application of the above theory to four selected regions, and the comparison of the observed and computed values. Four stations on the coast were selected in which the factors entering into the computation differed widely. For (ti) the value 8^ was used north of latitude 36^ and 9^ was used for the region south, but the value of (xi-j-yXo) was chosen so as to give the best agree- ment between the computed and observed values, but was assumed constant for each statien. It seemed reasonable to use a value of (ti) somewhat greater than the bottom temperature, as the water would become w^armer as it rose and mixed with the layers above. The value chosen is about the average of the mean annual surface temperature Ocean Temperatures along the West Coast of North America. 265 and the bottom temperature. It is the actual value usually found at the depth of 500 meters. Using for (V^y) the average wind velocity for the month's time just proceeding the middle of the month to which the mean tem- peratures (t2) and (T) correspond, the value of (t>) for the latitude, and a constant value of (xi-\- — X2\ determined by trial, the temperature difference (t) and the actual temperature (T) of the in- shore water were calculated for each month of the year. And the results were compared with the observed values. The values of (V^) were calculated^) from the U. S. Coast Pilot Chart records,^) and the temperatures were taken from Thorado's maps, on which the surface isotherms for each month of the year are plotted between the coast and longitude 140^, and between the latitudes 20^ and 50^ In the following tabulation of results San Diego is denoted by (1), Point Conception by (2), San Francisco by (3) and Cape Mendocino by (4). The same results are shown graphically by (Figs. 8, 9, 10 and 11). Station No. 1, Lat. 32<* 45', Long. 118^ / sin O'^ 0,736, ti = 9^ (xi + yX2) = 915 Kilometers, t = 0,017 V^(t2 — ti) Month V f. (t2 - tj calculated observed calculated observed Differen- '^ w t t T T ces 1 11,80 16,90 7,90 1,60 1,90 15,30 15,00 0,30 2 13,45 16,60 7,60 1,75 1,20 14,85 15,40 -0,55 3 14,10 15,80 6,80 1,65 0,20 14,15 15,60 -1,45 4 13,20 15,20 6,20 1,40 -0,10 13,80 15,30 -1,50 5 15,40 17,50 8.50 2,20 3,20 15 30 14,30 1,00 6 i6,yo 17,50 8,50 2,45 0,90 15,05 16.60 —1,55 7 18,20 20,90 11,90 3,70 4,90 17,20 16,00 1,20 8 18,40 21,90 12,90 4,00 3,90 17,90 18,00 —0,10 9 17.60 21,50 12,50 3,75 4,10 17,75 17,40 0,35 10 16,90 20,10 11,10 3,20 2,90 16,90 17,20 -0,30 11 15,55 18.70 9,70 2,55 2,20 16,15 16,50 -0.35 12 14,15 17,75 8,75 2,10 1.55 15,65 16,20 0,45 Average value of all of the observed values of t, 2,23 „ „ t, 4,30. „ the three greatest ^) The frequency, or number of hours per hundred during which the average wind velocity in each 5 degree square has the direction N, NNE, NE, etc. for each month is denoted by the length of an arrow pointing in the direction of the wind. The average force of the wind is expressed in Beaufort's scale. The velocity in miles per hour is given by the formula V = (5 F) -j- 3, where F is the force in Beaufort's scale. (See bulletin. Instructions to Marine Meteorological Observers of the U. S. Weather Bureau, by James Page for a table showing the relation of (F) Revue d.ges.Hydrobiol.u.Hydrogr. Bd.V. H. 2/3, Jg 266 G. F. McEvven. Station No. 2, Lat. 35^ Long. 121^ /sin

=0,792, ti (xi + ^ x,) = 440 Kilometers, t = 0,030 V,,. (t.2 — tj. Month Vw U {t2— ti) calculated observed calculated observed Differen- t t T T ces 1 7,70 14,20 6,20 1,40 1,70 i2,80 12,50 0,30 2 6,87 13,80 5.80 1,20 2.20 2,60 11,60 1,00 3 9,25 12,60 4,60 1,30 1,10 ^1,30 11,50 -0,20 4 11,40 12,00 4,00 1,40 0,70 l0,60 11,30 —0,70 5 12,50 14.50 6,50 2,40 3,20 12,10 11,30 0,80 6 14,40 15,20 7,20 3,10 1,40 12,10 13,80 —1,70 7 17,50 20,00 12,00 6,30 6,50 13,70 13,50 0,20 8 18,60 19,90 11,90 6,65 6,90 13,25 13,00 0,25 9 17,60 19,90 11,90 6,30 6,10 13,60 13,80 -0,20 10 13,60 18,60 10,60 4,30 4,00 14,30 14,60 -0,30 11 8,80 16,60 8,60 2,30 2,80 14,30 13,90 0,50 12 6.65 15.20 7,20 1,40 1 1,80 13,80 13,40 0,40 Average of all the observed values of t, 3,20. „ „ the three greatest „ „ t, 6,50. to (V) ). The average monthly value of the component (V^^.) parallel to the coast was found by resolving each velocity into directions perpendicular and parallel to the coast and adding the products of all the latter components by their corresponding frequencies and dividing this sum by the sum of the frequencies. ^) The values of (V^J were obtained only from the observations in the first square west of the corresponding station, tlie centers of the squares used being about 400 or 500 kilometers from land. Only values of (Vw) during the time from 1908 to 1911 were used. Ocean Temperatures along the West Coast of North America. 267 I I r I S3Jn4 PU 3 c/uj9j_ Cvj — OOoON-iOlO^T'O CM CNCNJCN— — _ — __— — ..^a 0 0»o;^cvjocouD';I-cmO cn T r\ O vn -^ HO CM — -■: y / -''" A 7 ..'■ ^ < ^,_ — - r--^ ^- ^— ^ y ^ =^ ,, ."■ "" ^ si=- j ^ ; ^ S \ ^ ^ ^^N ^ \ V ""■ — - *.^ >,^ ""■ ■^^^i K ~~-. N -., ^^ ^ N / ) \ \ / ) "" — - -, / y y / ,. .'' ( / '-' A '/ y' /9 .-' Jf ,-' ■ .y ^ ,,'"' \ S^ , ''' ^f "^^ / \ > (' «. ) }) \ k ^s ■-• •— f^ "■*» ., **5^ -^ V. ^ ^ ^'n / ^r ^ " — - '-- -v ^ \ V "^s i^ / ( / I \ s, y ,.-' '/ •-J ^N, .( if / // / J >^ ' /I /I 7 / ^ y ,'■ / / /i ^-"^ / , > S ' N \ s V . --. \ ^ N N "4 — -^ V ( U \ f-^ ^-^ "- .... :^ \ *N ^-^ -^^ r: /> 1 y ,■'' // '^ K /' ( \^ \ / ,. ' / Ji .--'■ ' > ^/ / / Jl" i^ • y J 1 fe \ ( . /\\ .... ^ > \, ^ -J^ K"- \ -J ^ '\ ■^ A- '• -- ....^ < ^^ „ * ^1^ fT"^ / ' ; k 1 / V _/ \, \ L- ^ o tc 00 268 G. P. McEwen. Station No. 4, Lai 40^ Long. 124^ V sin ^ = 0,802 t, = 8^ (xi + I xo) = 263 Kilometers, t = 0,0497 (V J (t, — W. Month *) ^w U {t.-ti) calculated observed calculated observed Differen- t t T T ces 1 —4,9 12,60 4,60 0,00 1,30 12,60 11.30 1,30 2 -6,40 11,80 3,80 0,00 1,10 11,80 10,70 1,10 3 -2,40 11,40 3,40 0,00 0.70 11,40 10,70 0,70 4 0,15 10,60 2,60 0,00 —0,40 10,60 11,00 -0,40 5 1,30 12,40 4,40 0,20 1,40 12,20 11,00 1.20 6 6,95 14,30 6,30 2,15 2,30 12.15 12,00 0,15 7 11.60 18,40 10,40 6,00 0,40 12,40 12,00 0,40 8 14,50 19,10 11,10 8,00 8,00 11,10 11,10 0,00 9 13,55 19,20 11,20 7.55 7.20 11,65 12,00 —0.35 10 8,35 17,70 9,70 4,10 3 70 13,60 14,00 —0,40 11 -1,85 15,40 7,40 0.00 2,40 15,40 13,00 2,40 12 —6,30 13,80 5,80 0,00 1,40 13.80 12,10 1,40 Average of all of the observed values of t, 2,98. „ „ the three greatest „ „ t, 7,20. VII. Discussion of the methods and results, and additional tests of the theory. The assumptions (page 257) on which the computation of the sur- face temperatures were based are not in accord with the actual con- ditions, first, because the direction of the coast is not north and south except at station (4), second, because the flow is not normal to the coast as assumed, but, as will be shown, is directed toward the south west at angles varying from (0^) to (90®) with the coast. To justify the method used in computing the temperatures, and to explain various peculiarities of temperature distribution, a more detailed consideration of the distribution and magnitude of the currents deduced from Ekman's theory will be required. But the necessary mathematics for such a quantitative study can be avoided and a qualitative explanation of some of the results can be worked out by assuming an ideal set of conditions and then estimating the change in the result due to such a modification of these assumed conditions as will secure an agreement with the actual circumstances observed at the different regions. If a steady wind uniform in direction and velocity blows parallel to a long straight coast, and the depth of the water is constant and *) The sign (— ) means that the component (V^^,) is directed to the north, in all other cases it is directed to the south. Ocean Temperatures along the AVest Coast of North America. 269 greater than (2 D), the actual velocity at any point of the surface current is the resultant of the velocity (Uo) (problem 2, page 253) and the velocity in a surface layer which the corresponding local wind would produce in an ocean extending to a great distance from the point in all directions, according to problem (1), page (251). But along the Pacific coast, the wind direction, except during a few weeks in the winter is toward the south east, and makes an angle of about (25^) with the coast within about (300) kilometers from the shore. Therefore the angle that the surface velocity (V) would make with the coast would be (45^) — (25^) = (20^) if (Uo) were zero, and it can be shown the actual value of (Uo) next to the coast reduces this to (10**). But the wind direction shifts to the west at points to the south and west of Cape Mendocino, and makes an angle of about (45^) to the west of a north and south line at the latitude of (32^) and longitude (140®) in the summer. Therefore, we would expect (Uo) to diminish and the surface velocity to be directed more and more to the west as the distance from the coast increases and the latitude diminishes. Therefore a stream-line (a line having at each point the direction of the motion at that point) starting from the vicinity of Cape Men- docino would be about parallel to the coast there, but would con- tinually turn to the right as the latitude diminished. A series of such stream-lines starting from points equidistant from each other along the coast would divide the water area into a series of sections, such that the water upwelling into each one would come from the ocean bottom underneath the part next to the coast, and no surface water could intrude from other regions. Observation shows that the cooling effect of the upwelling water diminishes as the distance from the coast along a stream-line increases, and in the following discussion, we will assume for the outer boundary of any section a line parallel to that part of the coast bounding the opposite end, and at such a distance that the temperature is practically normal. Now the cooling effect in an east and west area such as was considered on page (257) could be determined by computing the amount of water upwelling into each section that crosses it, and then calculating the temperature of the part of each section included in this area. From the appearance of the isotherms, the actual as well as the normal temperature of any section increases in proportion to the distance from the coast along the corresponding stream lines, therefore the assumptions stated on page (257) except as to direction of flow, corre- 270 G. F. McEwen. spond to the conditions in a section included between two stream-lines, and the method described on page (259) will give the value of (T), the actual temperature near the coast, if the mean value of the normal temperature were used for (t2). Thus it would be possible to obtain the value of (T) from the section that started where (T) was desired, and the temperature of an area running east and west could be found from the temperatures of the inclined sections that cross it. Now consider two of these inclined sections, one terminating at the outer boundary of the east and west area, and the other beginning at its inner boundary. Assume the outer boundaries of the three areas to be por- tions of a line parallel to the coast, then if the breadths of these areas mea- sured parallel to this line are equal, the areas will be equal, the mean temperature of the middle area will be approximately the average of the temperatures of the inclined sections, the mean wind velocity over the middle area will be approximately the average of the wind velo- cities over the inchned sections. Therefore if the normal temperature of the east and west area is assumed to be reduced only by the in- trusion of bottom water underneath it whose amount is computed as on page (259), where the wind velocity corresponding to that latitude is used, the result would be approximately the same as the more com- plicated but pertinent method just explained. Also the distance (xi + X2) is multiplied by the perpendicular distance between the bounding lines of the east and west surface to obtain the area, but the distance (x) should be multiplied by the length of the coast included by these bounding lines, since it is the quantity flowing normal to the coast that is assumed to enter the first area. Therefore the numerator of equation (13) and also of (22) should be divided by (cos«) where (a) is the angle the coast makes with a north and south line, and consequently the denominator (xi + yXg) deter- mined as on page (259) would be increased in the same proportion. Thus the use of formula (22) for computing temperatures is justified, but we must divide the quantity (xi-{--^X2\ determined as on page (259), by (cos a) in order to obtain the correct distance out to where the temperature is the average of (T) and (U) for the latitude. And if another relation between (xi) and (x^) can be found, then we can cal- culate (xi-f-Xa) the distance west to where the cooling effect is prac- tically negligible. Ocean Temperatures along the West Coast of North America. 271 Under the uniform conditions assumed on page (262), if the coast is vertical the upwelling would occur in a narrow belt as shown by (Pigs. 7 and 12). And the motion except within a distance from the coast of the same order of magnitude as (D), would be the same as that described on page (262), problem (3). If, instead of being level, the bottom slopes gradually upward toward the coast, the motion would be but little changed till the depth diminished from the value (2 D) (see note, page 255). From there toward the shore, the flow normal to the Fig. 12. Fig. 13. The arrows represent the average flow at the points 0. Cross sections normal to the coast. coast in the surface layer would continually diminish, and consequently the pressure gradient corresponding to the slope of the water surface and the flow toward the coast in the bottom layer would decrease as the coast was approached. If the depth is less than 0,5 (D), the flow is nearly parallel to the coast, and the surface is horizontal. Therefore (Fig. 13) the upwelling would begui where the depth is 2 (D) and would continue thruout the region from th-ere till the depth had de- creased to 0,5 (D), instead of being confined to a narrow region adjacent to the shore. 272 G. F. McEwen. That is, the greater the inclination of the bottom, in the neigh- borhood of a coast, or the more rapidly the depth increases with the distance from the coast, the narrower will be the belt near the coast into which a given volume of cold water intrudes, and therefore the greater will be the reduction of the surface temperature. For example, at the head of a submarine valley terminated at the coast, a lower surface temperature than that on either side would result from a wind so directed as to produce upwelling. Another factor affecting the upwelling is the steadiness of the winds. The effect of the winds over the broad and deep regions, re- quiring several months to become fully established, can not follow the seasonal change in the wind velocity, but tends to agree with that due to the average velocity^), and approaches more nearly to that average amount the more steady the wind. But the effect in narrower belts, and in the shallower water of the continental shelves, requiring but a few hours or days to become practically fully devoloped would change more nearly in accordance with the monthly variation of the wind velocity. We can now test the theory further by comparing the results of the general conclusions just reached with the corresponding observations. An estimate based upon the U. S. Coast Pilot Chart records of the breadth of the wind belt was made for each station and these distances were plotted as ordinates against distances from station (1) as abscissas (Fig. 14, curve 2). That is, this line which practically touches the coast at latitude (50®), (500) miles north of Cape Mendocino marks the appro- ximate western limit of winds whose velocities have a north and south component. Between the coast and this line the winds tend more and more to the west as the distance from the coast increases. Therefore the system of streamlines due to the observed wind velocities that would be expected from the theory would be limited to a narrow area corresponding to the narrow wind belt off Cape Mendocino but would spread out to the west and south, as the latitude diminished and the wind direction shifted to the west. Thus a general surface drift toward the south and bearing more and more to the west would ^) Owing to the large value of the effective internal friction of the water which tends to prevent the circulation, the motion would decrease nearly in accordance with the decrease of wind velocity, but would not increase to the maximum value which the wind velocity would generate if it continued for a sufficient length of time. Therefore, the actual effect of a varying wind would be less than that due to a steady wind of the same value as the average of the changing winds. Ocean Temperatures along the West Coast of North America. 273 result. This is in good agreement with observations^*^ (See also Figs. 20 and 21). Also the upwelling of cold water, which we have seen is confined to a narrow belt next to the coast and depends on the component 400 2200 2000 1800 1600 1400 1200 1000 800 600 400 200 — 0 7