MONTHLY WEATHER REVIEW Editor, ALFRED J. HENRY
OCTOBER, 1928 CLOSED DECEMBER 3, 1928
ISSUDD JANUARY 5. 1929 VOL. 56, No. 10
W. B. No. 969
~
AMOUNT OF SOLAR RADIATION THAT REACHES THE SURFACE OF THE EARTH ON
THE LAND AND ON THE SEA, AND METHODS BY WHICH 1% IS MEASURED'
HERBERT H. I<II\IBALL
[Weather Bureau, Washington, September 26,19281
The time available for presenting this paper will not permit a detailed discussion of all the points involved.
Therefore, for a description of pyrheliometers employed in radiation measurenients I would refer you to a paper
by Professor Marvin and myself on Solar Radiation and Weather Forecasting, in the Journal of the Franklin
Institute, volume 202, page 273, September, 1926.
It may be added that since the above paper was
written there has been published a description of impor- tant improvements which have been made in the Smith-
sonian silver disk pyrheliometer and which greatly decrease
the sky area to which the thermal eleinent is exposed. Also, thermo-electric pyrheliometers are finding increased
favor for use in obtaining continuous records of the total solar radiation (direct + diffuse) received on a horizontal
surface. The Smithsonian pyrheliometric scale of 1913 is the
nccepted standard in most countries, although the Anqstrom standard still has adherents. Recently a
radiogram from Prague, picked up by an amateur in this country, informs me that an absolute ice pyrhelio-
meter has been constructed by Professor Vol6sin of the
Karlova University in that city. Details have not yet
been received. I n the MONTHLY WEATHER REVIEW for April, 1927,
volume 55, page 155, I have given a summary of pyrhelio- metric measurements made a t about 100 different points,
nearly all of which are inland, and frequently a t consider- able altitudes above sea level. In response to a recent
circular I have learned of several additions and some
corrections to be made to this summary. Having thus disposed of the beginning and ending of my subject I will devote the remainder of my time to a
discussion of the amount of radiation that reaches the surface of the sea.
Very few pyrheliometric measurements have been made a t strictly marine stations. The list includes
measurements by Thonison, a t Apia, Samoa (1); Linke
a t sea between Hamburg, Germany, and Buenos Aires,
Argentina (2) ; Gorczyhski, a t sea between Antwerp and Bangkok (3) ; Westman, a t Treurenberg, Spitzbergen
(4); a t Cape Horn, Chile, during the International Polar
Expedition of 1882-83 (5) ; and an important measure- ment a t Flint Island, by Abbot, during a solar eclipse
expedition in 1907 (6).
1 This paper by H. H. Kimball and the following paper by ct. F. McEwen were pre- sented before the joint meeting of the sections of nieteorology and oceanography during the ninth annual assembly of the American Geophysical Union held 3t Washington. D. C., Apr. 26, 1928, in the building of the National Academy of Sciences. The joint meeting was devoted to a symposium and discussion on interrelations between the
888 and the atmosphere, and the effect of these relations on weather and climate. The communications presented were on problems related to (a) solar radiation, (b ) surface- water temperatures, and (c) atmospheric circulation. The two papers here printed were under (a); the other complete papers under this subhead, or references to wher,e they have been published, appear in Bulletin No. 68 of the National Research Council which contains also references to the communications presented under (6) and (e).
24136-28-1
These observat,ions by no means cover the seven seas,
so I have sought to determine if our'knowledge of
met,eorological condit,ions over the oceans, and of the relat,ion between meteorological condit.ions and solar radiation intensities at the surfac.e of the earth, is not
sufficient t,o enable us to compute mean solar radiation int,ensit,ies for dift'erent lat,itudes with reasonable accuracy.
Figure 1 is a chart for computing the transmission for
solar radiat,ion of dust-free air when its water vapor cont,ent is known. Using an equation developed by King (7) from Rayleigh's classical work (8) we may
compute the at,niospheric transmission, ax, for different
wave len@hs of light through pure dry air of any desired
barometric pressure. I have made these computations
for the 38 different wave lengths for which Abbot has given what he consideres the most reliable relative energy
intensities, lox, oubside the atmosphere (9). Then by Lambert's formula, a.Xm = u,? we may determine what
will be the form of the solar spectrum energy curve after
the solar rays have passed through pure dry air of a given pressure. I have made the computations for pressures of 40.0 and 76.0 cm. of mercury. Considering passage through the latter when the sun is in the'zenith to repre-
sent unit air mass, I have extended the computations to
air masses 2.0, 3.0, and 4.0, which represent solar zenith distances of 60", 70?7, and 75".
Effecting a graphic.al integrat,ion by finding the area
under these various curves, and applying Abbot's (10)
latest published correcbions for energy beyond the limits of his measurements, the pure dry air transmission, a'm,
for the total radiat,ion through the different air masses, is given by the ratio of the respective areas to that for
IoX. A smooth curve through these transmissions, which are plotted on their logarithmic scale as ordinates
against their air masses as absc.issas, gives curve (1).
Similarly, using Fowle 's (11) values of -the transmission
of water vapor for solar radiation, aWX, and disregarding selective absorption, we obtain the transmissions repre-
sented by curves (2) to (8), inclusive, Or, stated in another way, the difference between 1 .OO and the trans-
missions given by curves (1) to @), gives the depletion of solar radiation by scattering in passing through dust-
free air having a water-vapor content indicated by w,
and a length of path through the atmosphere given by
m. At sea. level w=2.3el where e. is the water-vapor pressure expressed in cm.*
To compute the depleton of solar radiation represented by the great water-vapor bands in hhe solar spectrum I have made use of curves given by Fowle [(12) Fig. 41,
and my computed values of l o h (a a h ~x ) for the values
2 This relation between w and e is true only for mean values of e for a considerable period. In dealing with individusl observations it may give results seriously in error.
393
394 MONTHLY WEATHER REVIEW OCTOBER, 1928
of X covered by the bands (12) [p. 4081. The plotted
results give curve (16), Figure 1.
Subtracting from the values given by curves (2) to (8), inclusive, the water vapor absorption for correspond-
ing values of wm given by curve (16) increased by 0.005
to take account of the selective absorption by the per- manent gases of the atmosphere (12), we obtain the
atmospheric transmission : a",, represented by curves (9)
to (15). These curves give atmoshperjc transmjssions
for dust-free air containing the amounts of precipitable water, w, indxated. Finally, Linke (13) has' defined atmospheric turbidity
as the number of clear dry atmospheres, which together bring about the same extinction of radiation as the actual
turbid moist atmosphere. He expresses it by the
equation - 1 1 log _o
-m log a', I, T=
where a'm is the atmospheric transmission for pure dry air through air mass m, and Io and I, are the solar radiation intensities a t air masses 0 and m, respectively.
Air Mass. m. Pressure - 763 rm)
u r n
FIG. 1.-Atmospheric transmission of solar radiation through dust-free air: Curves (1) to (8) after scattering by dust-free air containing the quantities of water-vapor indi- cated.by s; curves (9) to (15) after scattering and absorption by dust-free moist air contaming the quantities of water-vapor indicated by w: curve (16) absorption by water-vapor for different values of the product. u ~m
I have sought to compute the atmospheri? turbidity due to dust alone, Td, by substituting for a',,, in equation
(5) the atmospheric transmission for dust-free moist air,
a",, as given by curves (9) to (15), Figure 1.
A concrete example of the use of Figure 1 follows: At Apia, Samoa, Thomson (1) divides the year into
three seasons, as follows:
Dry season, May to August, inclusive. Equinoctial season, March, April, September, October.
Wet season, November to February, inclusive. Table 1 gives seasonal means of the meteoro1ogic.d
.elements with which we are particularly concerned.
TABLE 1.-Seasonal means for Apia, Samoa
Dry ....___________ 75.9 1.97 73.9 4.53 0.972 0.653 0.6% 0.424 Equinoctial ______ 75.8
1 $I33 1
i3.7
1
4.76
1
,970
I
.637
I
.520
I
.441 W-et _._.___________ 75.6 73.5 4.83 .967 ,674 .570 .475
From Table 1, Figure 1, and equation (5) the values of a" and Td given in Table 2 have been computed.
TABLE 2.--Beasonal values of a" and Td for Apia, Samoa
Dry ___._._......_._____ IO.6861 0.5541 0.4601 1.141 1 .0 6 1 1.11 1.03 Equinoctial .__...___.___ .&SO .547 ,451 1.18 1.08 Wet __________..__._____ .678 .W5 .448 1.02 .E 6 .927
For extreme accuracy we should take for unit air mass the ratios B-eJ76.0 given in Table 1. In the above example the value a" is thereby increased about 0.002,
which is inconsequential.
TABLE 3.-Daily totals of solar radiation (direct+diffuse) received
O )L a horizontal surface in the absence of clouds. Gr. cal. per cni'.
Lati- tude
Roo N.. 60" N..
56" N..
520 N..
48' N-.
42O N..
3G0 N-. 30" N..
200 N-. 10' N..
00- - - - -.
100 s--.
200 s--.
3OoS--. 36" S --.
42'S-.. 480 s.-. 52'6 ... 56O S.-. 600 s--.
.--______---
204 376 582 229 413 628 240 415 620 352 437 639 307 460 E41 314 472 656
409 536 686 335 514 6% 444 592 727 404 582 723 459 612 716 466 629 722 519 630 739 610 602 736 655 726 722
688 6% 6701
738 681 605
708 ae 54s 721 675 588
751 626 505 710 577 440
704 537 383 655 507 322 652 452 283 637 411 234 631 380 203
818 896 735 771 793 704 750 799 773 824 763 795 782 827 790 842 770 805 802 811 807 830 732 743 768 732 772 772 751 790 702 BB8
626 809
538 513
473 Ppa
496 497
386 354 327 282
2 8 219 202, 161
1w)/ 113 110
726 724 736 735 754 736 721 743 770 716 692 755 736 701
623
555
473
508
396 334
263 203 162 113
.-.-
- - - - - - - - 656 361
584 388 578 388 589 390 589 415 650 460 634 495 622 '474 639 514 674 522 653 546 €33 516 692 593 706 663 712 7 M
667 687
630 695
563 661 603 680
514 642 454 591
387 575 376 510
299 462 238 427 219 407
____________ 193 ____ ____ 208 - - - - - - -. 228 105 -... 235 115.-.. a87 164 119 304 171 123 328 214 179 313 202 180 380 285 258
391 270 233 433 341 317 424 388 300 479 420 383 587 487 408
645 574 553
685 657 850
7ia 718 m
?ai 787 733 818 7 ~8
712 850 893
731 830 867
735 818 888 684828900
ea3 830 883
652 808 819
647 Me 881
The values of T,, less than unity during the wet season,
while partly due to the fact that frequent showers keep the air nearly free from dust (l), are no doubt principally
due to the shallowness of the southeast trades a t this season (l), and, in consequence, an overestimate of the
value of 20.
Determinations of Td from observations on individual
days at sea give great variations in its value. From all the observations available I have been led to use the value
1.15 escept near the west coast of Africa, where Linke's observations indicate a very considerable increase in the
atmospheric dustiness, and duiing the wet season in the
tropics, when the value *O.O was employed. It is to be noted that Td usually decreases in value m t h increase in
air mass.
Atmospheric transmissions have been computed from Figure 1 for the latitudes indicated in Table 3 and
corrected by Td. Estimates of the water vapor content of the atmosphere have been based on monthly means of
temperature and relative humidity for about 140 stations,
obtained principally from the various volumes of the Pilot, published by the hydrographic department,
British Admiralty, and from nieteorological summaries
prepared by Reed (14). Vapor pressure values thus
OCTOBER, 1928 MONTHLY WEATHER REVIEW 395
obtained are undoubtedly more accurate for island than for continental stations, on account of the relatively
small temperature variations a t sea. The stations selected are distributed in latitude from
Treurenberg, Spitzbergen, 79' 55' N., to Laurie Island,
South Orlmeys, 60' 44' S. They have been grouped so as to give the vapor pressures a t the latitudes indicated
in Table 3. In the Tropics there is little variation in
vapor pressure with longitude. In temperate regions the west shores of oceans usually show higher humidities
than east shores, especially in north latitudes, probably on account of the warm west-shore ocean currents.
The vapor pressures have been grouped so as to bring
out these differences, which are reflected in the radiation intensities.
Atmospheric transnlissions give solar radiation inten- sities on the assumption that the value of the solar con-
stant is 1.0. They have been computed for each hour angle of the sun from noon for the dates indicated in Table 3, which have been selected so as to include the
dates of greatest north and south declination of the sun,
dates when its declination is, &O, and dates in July, August, October, and November, which have the same
solar declination as the 21st of May, April, February,
and January, respectively. They have been multiplied by the sine of the sun's altitude at each hour from noon,
apparent time, to obtain the relative intensity of the solar radiation on a horizontal surface, and increased by
a proportional part, depending upon the water-vapor
content of the atmosphere and the solar altitude, so as to include in the total the diffuse solar radiation from
the sky (15). These final values have then been plotted against intensities as ordinates and hour angles as abscis- sas and a smooth curve drawn through them from the
value a t noon to 0 a t the time of sunrise or sunset. A
graphical integration is effected by finding the area under this curve. The area is then multiplied by twice the
solar constant divided by the square of the earth's relative
solar distance on the dates indicated for each month.
The result given in Table 3 is the average radiation to be expected on the dates indicated with cloudless sky con-
ditions, expressed in gram-calories per cm? per day. It is to be noted that in both hemispheres the daily totals
average higher in the spiing months than in the fall. Figure 2 shows graphically the variations with latitude
in the total solar radiation received over the oceans on
the 21st of June. No radiation is then received south of the Antarctic Circle. There is in general a gradual
increase in the daily total until latitude 48' N. is reached,
although the changes are slight from latitude 42' N. to
the pole. Increasing length of day and decreasing water-
vapor content of the atmosphere unite to give maximum values a t high north latitudes, in spite of the low altitude of the sun even a t midday. The higher values on the
east shores of oceans than on the west is well brought out in the daily totals for latitude 20' S. and 30' N.
At latitude 48' N. the conditions are reversed over the
Atlantic Ocean on account of the cold Labrador current
on the west shore. On December 21 (fig. 3) daily maxima are reached a t about latitude 48' S. with little change to latitude 60' S. I n general, there is less variation in
the daily totals with longitude in the southern hemisphere than in the northern.
From the same sources as for temperature and humid- ity, and for about the same number of stations, monthly means of cloudiness have been obtained. Angstrom (16)
and others have determined an approximate relation between daily totals of solar radiation and both the duration of sunshine and the average cloudiness. The
latter relation is not so well determined as the former,
since the relation with cloudiness seems to vary with the percentage of cloudiness (17) and perhaps. also with
solar altitude (16). I have determined average daily totals of solar radiation, Q, from the totals with a cloud-
less sky, Qo, by the equation
Q=Q0(0.29+0.71~1.0- c]) (6 1
where C is the proportion of the sky covered by clouds,
and Qo is taken from Table 3. The results are given in Table 4, "Average daily totals of solar radiation received
on a horizontal surface.))
TABLE 4.-Average daily totals of solar radiation received on a horizontal surface (directSdiffuse), gr. ca1. per cm.2
i 356 367 322 __.. ____ 1 ____I ____ ____ 406 421 372 287 1891 399 365 345 244 1671 1021 ________
389 356 318 231 1831 110
397 424 422 364 284 165 108
92 ____ ____
372 389 328 229 182 107 49--..
54...-
378 383 354 288 212 140 80 54 3 W 357 326 304 188 131 74 63 78 414 405 388 335 255 153 Bo 71 449 477 421 376 317 235 153 120 240 313 435 486 524 442 382 318 233 151 129
260 334 380 389 321 360 375 275 242 208 188 247 365 420 4W2 441 432 399 3% 259 244 197 364 415 482 476 482 449 43i 420 340 301 281 225 303 353 376 332 378 372 296 249 216 189 420 466 475 452 493 464 450 428 379 324 307 404 437 440 383 404 412 424 4.09 384 350 345 483 556 527 477 414 422 439 444 453 403 382 385 427 394 363 340 348 348 359 338 321 314 361 348 328 297 230 269 283 326 349 344 336 385 411 432 448 428 433 511 531 520 480 362
395 402 403 355 337 366 397 419 408 363 355 471 421 356 309 294 350 411 458 469 453 464 455 451 406 381 380 397 451 483 444 432 439
428 389 353 305 285 308 363 417 465 508 488 .587 508 441 337 321 326 367 415 447 555 548
332 335 295 278 285 269 303 323 348 388 342 526 536 486 408 405 427 533 613 649 712 614 553 475 343 251 208 255 324 423 535 585 627 434 359 280 222 183 222 288 360 477 488 576 528 421 306 209 170 20.3 286 385 417 559 621 422 335 243 174 140 173 238 323 404 471 529
4Ck4 308 206 140 118 141 %38 289 395 4% 510 358 291 185 116 82 117 216 293 388 429 4E4 319 227 151 M) 52 78 155 232 333 406 439 320 198 113 55.--- 58 128 233 319 401 449 228 139 13 ____ ____ ____ 95 176 234 305 322
2 ~) 378 472 499 522 5x3 4 m 371 285 225 188
-
The variations in cloudiness with longitude cause much
greater variations in the daily totals of radiation than do the variations in absolute humidity. For this reason an
increased number of groupings along the different paral-
lels of latitude becomes necessary. These variations
become clearly apparent when the daily radiation totals
are charted as in Figures 4 to 7, inclusive. for December 21, June 21, March 21, and September 22.
The monthly mean cloudiness on the Gilbert Islands is
so much less than at other islands near the equator that
one may well ask if topographic features on islands may
not in some cases markedly modify the cloudiness, so
that it is not representative of the general cloudiness of its locality. Sufficient data was not at hand while this
paper was in prepa.r?tion to make a study of the topogra- phy of the different islands. It therefore seemed best to base the computations on the data as published.
-4 few solar radiat>ioii records are available for checking
the computed daily totals of solar radiation of Tables 3
and 4. Re.ference has already been made to measur? nients of the intensity of direct solar radiation at certam
niarine stations, and their use in computations of the
value of Td. It has also been found t,hat atmosphenc
396 MONTHLY WEATHER REVIEW OCTOBER, 1928
FIG ?.-Isopleths of total solarradiation (direct plus diffuse) on June 21 with cloudless sky (Qramalones per day per cm.: of horizontal surface)
Fro. 3.--Isopletlis of total solar rashtion (direct plus diffuse) on December 21 with cloudless sky (Gram-calories per day per cm.2 of horizontal surface)
FIO. 4.-Isopleths of total solar radiation (direct plus diffuse) on December 21 with average cloudiness (Qram-calories per day per cm.2 of horizontal surface)
OCTOBER, 1928 MONTHLY WEATHER REVIEW 397
FIO. 5.-Isopleths of total solar radiation (direct plus diffuse) on June 21 with average cloudiness I(Grarn6alories per day per cm.2 of horizontal surface)
FIG. tl.-Isopleths of total solar radiation (direct plus diffuse) on March 21 with average cloudiness (Oramcalories per day per cm.2 of horizontal surface)
FIG. 7.-Isopleths[of total solar radiation (direct plus diffuse) on September P with average cloudmess (Gramcalones per day per cm.2 of horizontal surface)
398 MONTHLY WEATHER REVIEW OCTOBER, 1928
transmission coefficients for Upsala, Sweden (18), lati-
tude 59’ 51’ N. axe, in general, only about 0.005 less than the transmissions I have computed for latitude
60’ N. Average daily totals of the radiation received on a
horizontal surface are available for Stockholm, Sweden,
latitude 59’ 21’ N., and for Habana, Cuba, latitude
23’ 09‘ N (18), both of which have a semimarine climate.
The records for Stockholm cover a single year, those for
Habana about 15 months. The Stockholm records give daily totals only a few per cent less than the computed
values of Table 4 for latitude 60’ N., except for May and
June. The published curve of daily totals [(18) Figure 21,
shows a decided depression for these two months. The
daily totals for Habana [(ls) Figure 11, are less than the
computed values of Table 4 for latitude 20’ N., for the
months October to February, inclusive, and more than the computed values for March to September, inclusive.
We would expect the total radiation a t Habana, latitude
23’ 09’ N., to be more than a t latitude 20’ N. in summer
and less in winter.
It seems evident that with reliable climatological data the radiation intensity over the oceans may be computed
with considerable accuracy;&u& the values here given must not be accepted a5 fi-nal. ---- __--
REFERENCES
(1) THOMSON, ANDREW.
(2) LINKE, F.
1927. Solar radiation observations at Apia, Samoa. Monthly Weather Review, 55 : 226.
1924. Results of measurements of solar radiation and atmospheric turbidity over the Atlantic Ocean and in Argentina. Monthly Weather Review,
52 : 157.
1923. Re ort of the Polish actinometric expedition to &am and the equatorial regions. Bulletin MBt6or- ologique, Institut Central MBtborologique 8. Var- sovie, Nr. 9-10, p. 90.
(4) WESTMAN, J. 1904. Mesures de l’intensit6 de la radiation solaire faites en 1899 et en 1900 B la station d’hivernage su6- doise B la baie de Treurenberg, Spitabergen. Mis- sion scientifique pour la mesure d’un arc de meridian au Spitzberg entprises en 1899-1902 sous les aus- pices des gouvernements Su6dois et Russe. Mission subdois.
1885. Observations des radiations solaires. Mission scien- tifique du Cap Horn, 1882-1883. Tome 2.
Meteorologie, Secunde partie, 8. 59-73, Paris.
1913. Annals Astrophysical Observatory, 111, 141-143.
Washington.
1913. On the scattering and absorption of light in gaseous media with applications t o the intensity of sky radiation. Phil. Trans. R. SOC. London. A.
212,375, London.
(3) GORCZY~~SKI, LADISLAUS.
(English summary.)
Tome 2, Section VIII, B. Stockholm.
(5) INTERNATIONAL POLAR EXPEDITION, 1882-1883 (France).
(6) ABBOT, C. G. and others.
(7) KINQ, LOUIS VESSAT.
(8) LORD RAYLEIQH. 1899. On the transmission of light through an atmosphere containing small particles in suspension, and on
the origin of the blue of the sky. Phil. Mag.
47,375. London.
(9) ABBOT, C. G. and others. 1923. The distribution of energy in the spectrum of the sun and stars. Smith. Misc. Coll. V. 74, No. 7.
Washington.
1927. Smithsonian solar radiation researches. Sonder- druck aus Gerland’s Beitriige aur Geophysik. Bd. 16, Heft 4, Leipzig.
1917. Water va or transparency to low temperature radia- tion. imithsonian Misc. Coll. Vol. 68, No. 8, Washington.
Astrophysical Journal, 42, 394.
(10) ABBOT, C. G.
(11) FOWLE, F. E., Jr.
(12) FOWLE, F. E., Jr. 1915. The transparency of aqueous vapor.
(13) LINKE, FRANZ, and BODA, KARL. 1922. Vorschlage zur Berechung des Trubungsgrades der AtmosphLre aus den Messungen der Intensitat der Sonnenstrahlung. (14) REED, W. W.
Monthly Weather Review, 54, 133, Washington. 1927.
Climatological data for the tropical islands of the Pacific Ocean (Oceana), Monthly Weather Review Supplement No. 28, washington. (15) KIMBALL, H. H.
1925. Records of total solar radiation and their relation to daylight intensity. Monthly Weather Review, 52 : 475, Table 1. (16) ANGSTROM, A.
1922. Note on the relation between time of sunshine and cloudiness in Stockholm, 1908-1920. Arkiv fur matematik, Astronomi och Fisik, Band 17, No. 15,
Stockholm.
1919. Variations in the total and luminous solar radiation with geographical position in the United States. Monthly Weather Review, 47 : 780.
1927. Measurements of solar radiation intensity and deter- minations of its depletion by the atmosphere, with bibliography of pyrheliometric observations. Monthly Weather Review, 55 : 155-169. Wmh- ington.
Met. Zeit, 39, 161.
1926. Climatological data for the West Indies.
(1’7) KIMBALL, HERBERT H.
(Figure 9.) (18) KIMBALL, HERBERT H.
HEATING AND COOLING OF WATER SURFACES’
By GEORGE F. MCEWEN
(Abstract)
Under this title a brief report was read based upon a
50-page manuscript entitled : “Mathematical Theory of Vertical Temperature Distribution in Water under the Action of Radiation, Evaporation, and the Resulting
Convection or Mising.” (Derivation of a general theory, and its illustration by means of numerical applic ations to
reservoirs, lakes, and oceans.) Since this manuscript will be ready in the summer of
1928 for publication in full as a bulletin (technical) of the
Scripps Institution of Oceanography, La Jolla, Calif., by
tbe University of California Press, only a brief abstract is presented here.
A mechanism has been devised involving the sinking of surface masses of water rendered relatively heavy, the
evaporation, conduction through the air, and back
radiation, and a compensating ascent of lighter, warmer masses. The mathematical theory of this mechanism of the downward diffusion of surface cooling led to a pair
of simultaneous differential equations involving tur- bulence, rate of surface cooling, rate at which solar radiation penetrates the surface, rate of vertical dis-
tribution of temperature and salinity, and rate of vertical
flow of the water. Methods have been worked out for applying these equations to numerical data, without the need of their
general solut,ion, which has not been attempted. Three
integrals appearing in these equations and depending upon the vertical variation of specific gravity have
been tabulated to facilitate numerical applications. Numerical re.sults hare been obtained by applying the
theory to serial temperatures in a tank of water, to serial temperatures of Lake Mendota, Wis., and to serial temperatures and salinities in the Pacific Ocean near
San Diego, Calif. From such observations the theory provided a means of e,stimating the rate of penetration of solar radiation through the water surface, the rate of
surface cooling, and, therefore, an approximate estimate
of evaporation and the rate of vertical flow of the water.
1 The full title as given by the author is “The rate at which solar radiatiun penetrates the surhce of lakes and oceans, and t.he rate at which the surffbce loses heat as deducad from serial tempersture~,bservations.”-Ed.