On the Origin of the Coloration of Seas and Lakes

Vas. Shuleikin.

At first glance at the blue surface of the sea on a clear summer day, the thought may arise that the coloration of this surface is caused by the reflection of the blue sky in the water. It is not difficult, however, to convince oneself that this is not quite so: for this it is enough merely to compare, on the one hand, the saturated blue color of the Nordkapp Current, which enters the waters of the Polar Sea, and, on the other hand, the pale, greenish-gray color of the Sea of Azov, over which a bright southern sky is blue.

It is therefore obvious that the coloration of the sea is influenced by certain internal causes, certain individual peculiarities.

The elucidation of these causes has occupied a whole series of investigators, beginning with Leonardo da Vinci; particularly thorough experimental work was carried out ten years ago by the German physicist O. von Aufsess.

Studying the coloration of the Bavarian lakes, he came to the conclusion that their visible color is caused by certain coloring substances mixed with the water and characteristic of each lake.

However, as the author of the present note succeeded in showing in 19211, such a consideration leads to erroneous results if, proceeding from it, one carries out a quantitative investigation.

On the contrary, the origin of the coloration of seas and lakes can be explained quite simply and in detail if one follows, step by step, the propagation of the light rays falling from above into the water and then somehow directed into the observer’s eye.

Let us suppose that on each square centimeter of the surface of the sea there falls from above a certain quantity of light energy \(J_0\), and that this incident light is perfectly white.

Let us divide the entire mass of seawater into thin horizontal layers. Since water possesses selective absorptive capacity, then—in the propagation of light from layer to layer—the energy of the rays

will be partly continuously absorbed, with the red rays being absorbed most strongly, and the violet and blue rays least strongly.

The deeper the light penetrates, the more intense and blue a coloration it acquires. But an observer looking at the sea from above would never see this blue light if in the layers of water there occurred only absorption of radiant energy and there were no causes forcing the rays of light to return upward.

In reality such a cause exists—there exists another process that takes place in the layers of water alongside the process of selective absorption. This cause is the scattering of light by the smallest particles always present in seawater in the form of a very fine suspension (the water molecules themselves also, of course, scatter light, but the effect produced by them is negligible in comparison with the effect produced by other, more active particles—aggregates of gas molecules dissolved in the water, etc. This circumstance is not taken into account, among other things, by the recently proposed theory of the origin of the coloration of seas, put forward by the Calcutta physicist Raman (C. V. Raman, 1922).

The laws of scattering of light by “turbid media” were thoroughly studied by Strutt (Lord Rayleigh), who showed (1871 and 1899) that the amount of light energy scattered by an infinitely thin layer \((dz)\) of a turbid medium is directly proportional to the intensity of the incident light \(J\), to the thickness \(dz\), and inversely proportional to the fourth power of the wavelength of the corresponding ray, i.e.:

\[ dJ=-\frac{a}{\lambda^{4}}\cdot J\cdot dz. \]

Here \(\lambda\) is the wavelength, \(a\) is a certain coefficient, which we shall henceforth call the coefficient of scattering; this coefficient depends on the concentration and sizes of the suspended particles1.

The complete attenuation of light by a layer of thickness \(dz\) is not difficult to find by adding the amount of light scattered and absorbed. The latter cannot be expressed analytically, but it can be represented in the form of a curve (No. 1 in Figure 1): wavelengths are plotted along the abscissa axis, and the corresponding absorption coefficients along the ordinate axis. Let us denote the latter by the symbol \(f(\lambda)\).

Studying the propagation of light from layer to layer, one can calculate the amount of energy that is scattered upward by each infinitely thin layer lying at some depth below the surface of the sea. Before such rays cast upward reach the surface and enter the observer’s eye, they must again pass through more

or a less thick layer of water, but this circumstance too is not difficult to take into account, if the laws of absorption and scattering of light in seawater are known.

Of course, the amount of energy sent upward by the various layers will not be the same: the deep layers will send incomparably less of it than the overlying layers. Summing the energy that has reached the surface of the sea from below, the author of the present note finally obtained the following expression:

\[ J=J_0\, \frac{\dfrac{1}{4}\dfrac{a}{\lambda^4}} {\dfrac{1}{4}\dfrac{a}{\lambda^4}+f(\lambda)}. \tag{1'} \]

Fig. 1.

By the letter \(J\) is denoted here the quantity of radiant energy emerging back upward through a unit of the sea surface; all the other symbols retain the meaning indicated above.

It is not difficult to see that \(J\) will be less than \(J_0\), and that the diminution will be unequal for different wavelengths \(\lambda\): thus, red rays will be weakened considerably more than blue ones.

Let us consider the quantities \(J_0\) and \(J\) somewhat more closely. The first of them—the flux of energy penetrating a unit surface from above downward—may be divided into two components. In fact, if the sea is illuminated from above: 1) by the solar disk and 2) by the celestial vault, and if the brightness of the illumination by the sun alone is equal to \(S_0\), and by the celestial vault alone to \(H_0\), then it is obvious that \(J_0=S_0+H_0\).

As for the brightness of the illumination of the sea surface from within by the colored light emerging from the depths, it is not difficult to connect it with the quantity of energy \(J\) emerging upward through a unit surface.

Indeed, since this light is diffuse, it is obvious that the brightness of illumination \(M_0\) is expressed as:

\[ M_0=\frac{J}{\pi}. \]

Substituting into equation \((1')\), in place of \(J_0\) and \(J\), their expressions, we find:

\[ M_0=\frac{S_0+H_0}{\pi}\cdot \frac{\dfrac{1}{4}\dfrac{a}{\lambda^4}} {\dfrac{1}{4}\dfrac{a}{\lambda^4}+f(\lambda)} \]

or, otherwise:

\[ \frac{M_0}{S_0+H_0} = \frac{1}{\pi}\cdot \frac{\dfrac{1}{4}\dfrac{a}{\lambda^4}} {\dfrac{1}{4}\dfrac{a}{\lambda^4}+f(\lambda)} \tag{1} \]

It is not difficult to see that the equality obtained makes it possible to calculate the spectrum of that internal light which imparts to the sea its characteristic coloration; for this it is necessary only to know \(f(\lambda)\) and \(a\).

The values of \(f(\lambda)\) for various wavelengths have been obtained by a whole series of experimenters. We borrow them from O. von Aufsess’s book Die Physikalischen Eigenschaften der Seen (Fig. 1, curve No. 1).

Fig. 2. (Depths of disappearance are expressed in meters.)

As for the value of \(a\), it too can be determined from experiment by finding the coefficient of “absorption” of light in sea water. It is not difficult to show that such an absorption coefficient \(m\) in fact consists of two parts: one depending on the absorption of light by water (in the true sense of the word), and the other depending on the scattering of light according to Lord Rayleigh’s law. Namely:

\[ m=f(\lambda)+\frac{1}{2}\frac{a}{\lambda^4}. \]

Determining \(m\) by means of the examination of water samples, one can find \(a\) (in Figure 1 there is shown, for example, curve \(m\) No. 5, obtained by Aufsess for the water of Lake Staffel and attributed by him to a special “coloring substance” dissolved in the water). But the quantity \(a\) can also be approximately determined by means of a simpler procedure: by observing the depth at which the so-called “Secchi disk” disappears in the water. It can be shown that the relation between the coefficient \(a\) and the depth of disappearance of the Secchi disk \(Z\) is expressed by the curve shown in Figure 21.

The depths at which the Secchi disk disappears have been determined by all oceanographic expeditions.

There is therefore reason to suppose that, for the waters of three of our typical seas—the Black, the Baltic, and the White—the coefficient \(a\) is, respectively:

\[ \begin{aligned} a&=0.004,\\ a&=0.020,\\ a&=0.070. \end{aligned} \]

These three values of \(a\) correspond precisely to the curves

\[ m=f(\lambda)+\frac{1}{2}\frac{a}{\lambda^{4}}, \]

plotted in Figure 1 under the numbers 2, 3, and 4.

Substituting the values of \(a\) into formula 1, we shall find the spectra of the internal light that produces the coloration of the Black Sea (spectrum No. 6), the Baltic Sea (spectrum No. 7), and the White Sea (spectrum No. 8).

It is not difficult to see that the spectrum of the Black Sea corresponds to reality: the relative brightness of the rays in the blue part of the spectrum is incomparably greater than in all the others; the sea is distinguished by a saturated dark-blue color. It is also evident that the coloration of the Baltic Sea will be lighter and greener than that of the Black Sea, while the coloration of the White Sea will be still more whitish and greenish.

Up to now we have assumed that the particles suspended in the water only scatter light, without absorbing it.

In fact, in natural waters there are often also particles of another kind—particles possessing the ability to absorb light, and moreover selectively. Such, for example, are particles of disturbed finely divided bottom material, plankton, and so forth.

Taking into account the combined action of particles of both kinds, one may obtain the spectrum of the internal light in the most general form:

\[ \frac{M_{0}}{S_{0}+H_{0}} = \frac{1}{\pi}\, \frac{ (1-\beta)\,\frac{1}{4}\frac{a}{\lambda^{4}} + \beta\cdot\frac{1}{2}\varphi(\lambda) }{ (1-\beta)\,\frac{1}{4}\frac{a}{\lambda^{4}} + f(\lambda) + \beta\left[1-\frac{1}{2}\varphi(\lambda)\right] } \ldots \quad (2) \]

Here \(\beta\) denotes the probability of a light ray encountering (over a path of 1 meter) a particle of the second kind; \(\varphi(\lambda)\) is the coefficient of selective reflection for the latter.

Calculations by formula (2) lead to the spectral curves shown in Figure 3.

They correspond to the case when particles of alumina are suspended in the water.

Curve 9 corresponds to a small concentration of the latter \((\beta=0.02)\); the visible color of the water is evidently greenish.

Curve 10 corresponds to a somewhat greater concentration \((\beta=0.20)\); the color is completely green.

Finally, curve 11 corresponds to the limiting case when $\beta = 1$, that is, when the scattering of light by particles of the first kind may be neglected; the visible color here is already brown, such as that of mountain streams carrying a large quantity of particles of clayey earth.

Thus, all the nuances of the coloration of bodies of water can be explained (and, moreover, quantitatively) if one traces the scattering and absorption of light in water.

This coloration depends, consequently, on the optical properties of the water of the given sea or lake; but its optical properties are, obviously, subject to changes.

Especially sharp changes are introduced by particles of the “second kind,” which appear in large quantity, especially after fresh winds, when from the shores and shoals whole bands of water often stretch out into the open sea, colored yellow-green or reddish (depending on the soil); colored plankton, which at times appears in very large quantities, acts in the same way.

But the number of particles of the “first kind” is also subject to change; apparently, their sizes too are not constant.

Fig. 3.

Indeed, in the water of seas and lakes, as is known, gases entering into the composition of atmospheric air are dissolved; thus, according to the data of Yu. M. Shokalsky, one liter of ocean water (salinity—35‰), at normal pressure and at a temperature of $15^\circ C$, absorbs $5.84\ \mathrm{cm}^3$ of oxygen and $11.12\ \mathrm{cm}^3$ of nitrogen. When the temperature is lowered to $0^\circ$, the amount of absorbed oxygen increases to $8.03\ \mathrm{cm}^3$, and the amount of nitrogen to $14.4\ \mathrm{cm}^3$; conversely, when the temperature of the water is raised to $30^\circ$, the amount of oxygen falls to $4.5\ \mathrm{cm}^3$, and the amount of nitrogen to $9.26\ \mathrm{cm}^3$.

Taking these figures into account, let us suppose that some cause or other brings about a local heating of cold sea or lake water. It is obvious that, in this case, part of the dissolved gases must be released; but the process of release of the gases in the form of more or less large bubbles will be preceded by a process of accumulation of gas molecules into groups, the number and sizes of which continuously increase with the rise in the temperature of the water.

These groups of molecules, apparently, are the very particles “of the first kind” mentioned above (“grouping” of gas molecules can, of course, occur even in the case when the gases do not even saturate the water at the given temperature).

It has become obvious, therefore, that with local heating of the waters of a sea or lake, the scattering coefficient must increase, and the visible coloration in that place becomes more whitish and greenish.

And indeed, the inhabitants of the shores of Lake Baikal observe whitish spots on its surface precisely in those places where hot springs beat.

The opposite effect takes place in the case when warm waters undergo local cooling. In this case the dimensions and number of molecular “clusters” must decrease, thereby decreasing the scattering coefficient itself and increasing the transparency of the waters. The latter will in such places assume a bluer, more saturated coloration, which is in fact observed, for example, in the Norwegian Current, where the warm waters of the Gulf Stream flow into the cold waters of the polar seas. Speaking of the visible coloration of the sea, we have hitherto assumed that the observer looks vertically downward—perpendicularly to the water surface—since only in this case can he observe the internal diffuse light in its purest form.

In reality, as Fresnel already showed, when light rays pass through the boundary of two dissimilar media (for example, water and air), part of the rays, being refracted, enters the second medium, while part is reflected back into the first. The greatest amount of radiant energy emerging into the second medium will occur in the case when the rays are directed normally to the boundary surface. But it is obvious that, besides the internal light, the observer also sees the reflection of the vault of heaven in the water, and this reflected light will have a minimum of brightness precisely with vertical observation.

Let us now suppose that the observer looks from the side of a ship (or from a sheer cliff over a very deep place) at the smooth surface of the sea extending to the horizon. It is obvious that the angle with the surface of the water at which he sees ever more distant parts of it will become sharper and sharper; therefore the visible brightness of the internal light will become less and less, and, conversely, the brightness of the reflected light will become greater and greater.

Directly below, under his feet, the observer will see a richly colored surface, while farther away it will become lighter and lighter, acquiring toward the horizon the whitish tint characteristic of calm weather...

On the basis of the laws of refraction and reflection of light, it is just as easy to explain the sharp changes in the visible coloration of the sea during agitation. Suppose that the ray of vision meets a certain element of an agitated water surface at the point \(O\) (Fig. 4), making an angle \(\varphi\) with the normal.

Obviously, the ray entering the observer’s eye can be resolved into two component rays: one of them \((M)\) emerged, after refraction, from beneath the surface of the water; the other \((H)\) was reflected from an element of the water surface, having fallen upon it in the direction \(H_0O\). Both the brightness of the refracted light \(M\) and the brightness of the reflected light \(H\) can be calculated by the Fresnel formulas, substituting in them the values \(M_0\) and \(H_0\). Namely:

\[ \left. \begin{aligned} M&=\frac{1}{2}\frac{\sin 2\varphi \sin 2\Psi}{\sin^2(\varphi+\Psi)} \left[1+\frac{1}{\cos^2(\varphi-\Psi)}\right]\cdot M_0,\\ H&=\frac{1}{2}\frac{\sin^2(\varphi-\Psi)}{\sin^2(\varphi+\Psi)} \left[1+\frac{\cos^2(\varphi+\Psi)}{\cos^2(\varphi-\Psi)}\right]\cdot H_0 \end{aligned} \right\} \ldots\ldots\ldots\ldots\ldots (3) \]

According to the observations of Bouguer, Roscoe, Perntner, and others, on clear days one may often take \(H_0=S_0\); thus, by means of equality (1) or (2), one can relate \(M_0\) and \(H_0\).

It is not difficult to see that then the system of equations (3) will make it possible to find the relative magnitude of \(M\) and \(H\) for any angles of inclination of the wave. In the last system of equations, the independent variable is, obviously, only one angle (for example \(\varphi\)), since the other angle is connected with it by Descartes’ relation:

\[ \frac{\sin\varphi}{\sin\Psi}=n=1.32 \]

(for the passage of a ray from water into air).

Fig. 4.

The changes of \(M\) and \(H\) can be represented very clearly on a polar diagram (see Fig. 5).

From the polar axis \(OA\) we shall lay off (counterclockwise) the angles \(AOC\) which the visual ray \(OA\) forms with the tangent to the wave at the given point \(O\).

Laying off on the radius vector \(OC\) the magnitude \(H\), we obtain curve I, expressing the visible brightness of the reflected light.

Laying off on it the magnitude \(M\) (on the same scale), we obtain curve II, expressing the visible brightness of the internal light in the blue part of the spectrum. Using equations (1) or (2), or else directly the spectral curves of Figures 1 and 3, one can construct the same kind of curve for any spectral rays emerging from the depths of the sea. For example, on diagram 5 one more of such curves (III) is plotted, calculated for the yellow-green part of the spectrum.

The polar diagram (see Fig. 5) makes it possible to follow the change in the color of the sea during wave motion.

Let the observer stand on the shore and look at the sea at an acute angle to the horizon. If the sea is calm, then the vector \(OC\) makes a very small angle with the polar axis \(OA\). The radius vector \((M)\) of curve II is close to zero, while the corresponding radius vector \((H)\) of curve I considerably exceeds it, approaching the maximum value \(OA=H_0\).

The color of the sea corresponds to the coloring of the firmament, usually whitish.

But as soon as waves form on the surface of the sea (even small ones), the angle \(AOC\) will begin to increase; at the same time the vector \(M\) will rapidly increase, and the vector \(H\) will sharply decrease; the sea will rapidly begin to turn blue (if it is the Black Sea) or else green (if it is the White or Baltic Sea).

Fig. 5.

In constructing diagram 5 it was assumed that the brightness of illumination of the sea by the firmament alone is equal to the brightness of illumination by the disk of the sun alone:

\[ H_0=S'_0. \]

But it is obvious that in reality \(S'_0\) may prove to be a very small quantity, close to zero, as occurs in the case when the disk of the sun is covered by clouds.

In this case, in formula (1) or (2), instead of \(S_0+H_0\) there will be only \(H_0\)—the coloration of the sea will be determined by the coloration of the firmament, all the more so because the vector \(M\), in most cases, will be small in comparison with the vector \(H\) (its relative magnitude will decrease approximately by half in comparison with the corresponding value in clear weather). This explains the dark, leaden color of the sea in cloudy weather. It is likewise not difficult to explain the peculiar, mottled coloration,

which, for example, the Black Sea takes on after a storm, when the sun in places peeps through the clouds: where the sun’s rays fall directly into the water, they are rapidly scattered by the large air bubbles suspended in the upper layers of the agitated water; being reflected upward by the latter, they pass through a comparatively small layer of water and here produce a yellowish-green coloration of the surface of the waves; on the contrary, where the clouds screen the direct rays of the sun, the observer sees a blue internal light issuing from the depths; here saturated-blue “shadows” of the clouds run along the waves.

The same diagram 5 makes it possible to explain also the intense coloration of mountain lakes. Indeed, the mountains surrounding the latter, as it were, “cut off” the belts of the celestial vault lying close to the horizon and reflected in the water at an acute angle. In the lake, consequently, only those zones will be reflected which correspond to large angles \(AOC\) (diagram 5), but the brightness of the reflected color will here be small, and the observer will see the coloration of the lake caused chiefly by the “internal” light.

This coloration will be either blue or green—depending on the optical properties of the lake water (see Figs. 1–3).

The quantitative conclusions of the theory set forth above were subjected by the author of the present note to experimental verification.

Initially the verification was carried out on a laboratory scale, at the Biophysical Institute, and complete agreement of theory with experiment was found. In addition to what has already been published in the Proceedings of the Physical Institute1, I shall give the result of a photometric observation of a “model” of the sea.

Into an aqueous solution of the blue aniline dye “rhoduline” there was poured an alcoholic solution of rosin, which produced in the water a very fine colloidal suspension.

By determining experimentally \(f(\lambda)\) for the solution of the blue dye (replacing the selective absorption in a thick layer of sea water) and the coefficient of light scattering by the finest particles of rosin (replacing particles of the “first kind” in sea water), it was possible to calculate the spectrum of the light issuing from a trough with parallel glass walls, if this trough, filled with the dye solution and the colloidal suspension, is viewed in reflected light. The calculation, performed by formula (1), leads to the curve shown in Figure 6.

Direct photometry of the illuminated surface of such a “model of the sea” gave a number of points plotted in Fig. 6. As is evident from the figure, the experimentally obtained points lie very close to the theoretical curve.

Fig.

Quite satisfactory results were also obtained in checking formula (2), with the role of particles of the “second kind” being played by ochre particles suspended in water.

The final verification of the theory under natural conditions was carried out last summer, in 1922. The first experiments were made in the Black Sea, near Sevastopol, on the harbor boat Ai-Foka, and gave results consistent with the theory. Unfortunately, owing to fresh weather and the strong rolling of the boat, the observations were greatly hampered, and there could be no question of constructing an exact spectral curve.

Under incomparably more favorable conditions it proved possible to carry out photometric measurements aboard the hydrographic vessel Pakhtusov, on which I was given the opportunity to work thanks to the kind assistance of the committee of the Floating Marine Scientific Institute and of the head of the Hydrographic Administration of the Northern Seas, P. P. Mikhailov.

The Pakhtusov left Arkhangelsk at the end of August and made a voyage to the shores of the Yamal Peninsula: through the White Sea, the Barents Sea north of Kolguev Island, the Yugorsky Shar, and the Kara Sea, with a call at the northern extremity of Vaigach Island—in the Kara Gates.

We present the results of observations at four of the stations.

\[ \frac{M_{0}}{H_{0}+S_{0}} \]

\(Z \to 1\)

\(0.2\)

\(0.1\)

\(0\)

\(0.64 \qquad 0.60 \qquad 0.56 \qquad 0.52\)

\[ \lambda \]

Legend:

\[ \begin{aligned} N1&\;—\;\circ\\ N2&\;—\;\circ\\ N4&\;—\;\bullet\\ N5&\;—\;\circ \end{aligned} \]

Fig. 7.

The spectral curves obtained at these stations are shown in Fig. 7.

As we see, their course fully agrees with the theory. Only a difference in scale is observed: the absolute brightness of the “internal” light proves to be less than that given by the theory, but this deviation is explained quite simply. Indeed, in deriving formulas (1) and (2) it was assumed that the concentration of suspended particles is constant both in the surface and in the deep layers of sea water. In reality this, of course, is not so: the scattering of light at different depths may be different—deep layers of water, in all probability, are more transparent than the surface layers. This assumption is also supported by observations of the coloration of deep-water plants, made by the author during the voyage on the Pakhtusov, but we cannot dwell here on these observations.

1. I. F. I. vol. II, 129, 1922. ↩↩↩↩