The Oceans Their Physics, Chemistry, and General Biology "d3_9

Eddy Conductivity, Diffusivity, and Viscosity

In the preceding discussion it has been repeatedly stated that the coefficients of heat conduction, diffusion, and viscosity that have been dealt with so far are applicable only if the water is at rest or in laminar flow. By laminar flow is understood a state in which sheets (laminae) of liquids move in an orderly manner such that random local fluctuations of velocity do not occur. However, the molecules of the liquid, including those of dissolved substances, move at random, and, owing to this

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random motion, an exchange of molecules takes place between adjacent layers. Consequently, there is a transfer of heat if adjacent layers are at different temperatures, a diffusion of dissolved substances if their concentrations are variable in space, or a transfer of momentum if their velocities differ. The amounts that are transferred are proportional to the gradients of temperature, concentration, or velocity, and the factors of proportionality—that is, the coefficients of thermal conductivity, diffusion, and viscosity—are among the characteristic properties of the liquid. For a given liquid they are functions of temperature, pressure, and concentration, and can be exactly determined in the laboratory.

In nature, laminar flow is rarely or never encountered, but, instead, turbulent flow, or turbulence, prevails. By turbulent flow is understood a state in which random motion of smaller or larger masses of the fluid is superimposed upon some simple pattern of flow. The character of the turbulence depends upon a number of factors, such as the average velocity of the flow, the average velocity gradients, and the boundaries of the system. Under these conditions the exchange between adjacent moving layers is not limited to the interchange of molecules, but masses of different dimensions also pass from one layer to another, carrying with them their characteristic properties. As a consequence, a snapshot of the instantaneous distribution of velocity, temperature, salinity, and other variables in the sea would show a most complicated pattern, but so far no means have been developed for establishing this picture. Measurements by sensitive current meters have demonstrated that in a given locality the velocity fluctuates from second to second, but in most cases observations of ocean currents give information as to mean velocities for time intervals that may vary from a few minutes to twenty-four hours or more. Similarly, special measurements have demonstrated that the details of the temperature distribution are very complicated, but in general observations are made at such great distances apart that only the major features of the temperature distribution are obtained. Inasmuch as it is impossible to observe the instantaneous distribution in space of temperature, salinity, and velocity, it follows that the corresponding gradients cannot be determined and that no basis exists for application to the processes in the sea of the coefficients of thermal conductivity, diffusion, and viscosity that have been determined in the laboratory. Since only certain average gradients can be determined, another approach has to be made when dealing with the processes in the sea. In order to illustrate this approach, let us first consider the viscosity.

In the case of laminar flow the coefficient of viscosity, μ, is defined by the equation τ_s = μdv/dn where τ_s is the shearing stress exerted on a surface of unit area, and dv/dn is the shear normal to that surface.

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In the case of turbulent flow a coefficient of eddy viscosity, A, can be defined in a similar manner:

[Equation]

where dū/dn now represents the shear of the observed velocities. The numerical value of the eddy viscosity depends upon the size and intensity of the eddies—that is, on the magnitude of the exchange of masses between adjacent layers; and the symbol A that is commonly used is an abbreviation for the term “ Austausch” introduced by Schmidt (1917). The numerical value of A also depends upon how the “average” velocities have been determined; that is, upon the distribution in space of the observations and upon the length of the time intervals to which the averages refer.

The definition of the eddy viscosity in the above manner appears purely formalistic, but it is based on the concept that masses which leave one layer carry with them the momentum corresponding to the average velocity in that layer, and that by impact they attain the momentum corresponding to the average velocity of their new surroundings before again leaving them (p. 472). Thus, A is an expression for the transfer of momentum of mean motion. This transfer is much increased by the turbulence, as is evident from the fact that the eddy viscosity is many times greater than the molecular viscosity.

The eddy viscosity can be determined only by examination of the effect on the mean motion. This effect is discussed on pp. 492 and 577, but a few points will be mentioned here. It has been found practical to distinguish between two types of turbulence in the sea—vertical and lateral, In the case of vertical turbulence the effective exchange of masses is related to comparatively slight random motion in a vertical direction or, if the term “eddy motion” is used, to small eddies in a vertical plane. Actually, the eddies are oriented at random, but only their vertical components produce any effect on the mean motion. The corresponding eddy viscosity has been found to vary between 1 and 1000 c.g.s. units, thus being one thousand to one million times greater than the molecular viscosity of water. In the case of lateral turbulence the effective exchange of masses is due to the existence of large quasi-horizontal eddies, The corresponding eddy viscosity depends upon dimensions of the system under consideration and has been found to vary between 10⁶ and 10⁸ c.g.s. units.

The distinction between vertical and lateral turbulence is particularly significant where the density of the water increases with depth, because such an increase influences the two types of turbulence in a different manner. Where the density of the sea water increases with depth (disregarding the effect of pressure), vertical random motion is impeded by

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Archimedean forces, because a mass which is brought to a higher level will be surrounded by water of less density and will tend to sink back to the level from which it came, and, similarly, a water mass moving downward will be surrounded by denser water and will tend to rise. In this case the stratification of the water is called stable, because it cannot be altered unless work against gravity is performed. Stable stratification reduces the vertical turbulence; where the stability is very great the vertical turbulence may become nearly suppressed and the eddy viscosity small. The effect of stability on the lateral turbulence, on the other hand, is negligible, because the lateral random motion takes place mainly along surfaces of equal density.

With regard to the eddy conductivity, similar reasoning is applicable. When dealing with eddy viscosity it was assumed that the exchange of mass leads to a transfer of momentum from one layer to another, which is expressed by means of A. Correspondingly, when dealing with eddy conductivity, one can assume that the transfer of heat through any surface is proportional to the exchange of mass through the surface, as expressed by A, and to the gradient of the observed temperatures, −d formula [Equation]/dn; that is, dQ/dt = −rA d formula [Equation]/dn, where r is a factor that depends upon the specific heat of the fluid and upon the manner in which the heat contents of the moving masses are given off to the surroundings. When dealing with homogeneous water, it is assumed that a large mass which is transferred to a new level breaks down at that level into smaller and smaller elements, and that equalization of temperature ultimately takes place by molecular heat conduction between the small elements and the surroundings. If such is the case, both the difference in momentum and the difference in heat content are leveled off, and the proportionality factor, r, is equal to the specific heat of the liquid. Since the specific heat of water is nearly unity, the numerical values of eddy conductivity and eddy viscosity are practically equal. However, where stable stratification prevails, the elements, being lighter or heavier than their surroundings, may return to their original level before completion of temperature equalization, but equalization of momentum may have been accomplished by collision. In this case, the factor of proportionality, r, will be smaller than the specific heat of the liquid; that is, in the sea, r is smaller than unity, and the eddy conductivity is smaller than the eddy viscosity. Thus, stable stratification reduces the vertical eddy conductivity even more than it reduces the vertical eddy viscosity. Taylor (1931) has presented the above reasoning in mathematical language (p. 476).

The discussion has so far been limited to a consideration of the vertical eddy conductivity, but lateral eddy conductivity due to lateral turbulence has also to be introduced. The numerical value of lateral

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eddy conductivity must be nearly equal to that of the lateral eddy viscosity, because the lateral turbulence is not affected by stable stratification.

Numerical values of the coefficients of eddy conductivity can be derived only from a study of the effect of mixing processes on the observed distribution of temperature. Methods of such determinations and numerical values are presented on pp. 483 and 484. The results have confirmed the above conclusions and have also demonstrated that the eddy conductivity varies within wide limits. In the upper layers of the sea, where stable stratification prevails, the vertical coefficient of eddy viscosity varies between 1 and 1000, whereas the corresponding eddy conductivity is smaller and varies between 0.01 and 100; in homogeneous water, however, no difference has been established (p. 485). In the cases in which the lateral coefficients of viscosity and conductivity have been examined, nearly equal numerical values have been found in agreement with the conclusion that the stability of the stratification does not influence the lateral turbulence.

The transfer of salinity or other concentration is similar to the heat transfer. The eddy diffusivity is also proportional to the exchange of mass as expressed by A, the factor of proportionality being a pure number. In sea water of uniform density r = 1, but in the case of stable stratification, when complete equalization of concentration does not take place, r < 1; that is, the vertical eddy diffusivity is smaller than A and equals the eddy conductivity. This conclusion has also been confirmed by observations (p. 484).