Conclusions as to Currents on the Basis of Tonguelike Distribution of Properties

Where horizontal differences in density are so small that differences in computed anomalies of distances between isobaric surfaces become doubtful, conclusions as to currents are often drawn from tonguelike distribution of properties, especially of temperature and salinity. In such cases, it is necessary to distinguish between tongues in a horizontal plane and those in a vertical section.

As a rule a tonguelike distribution in a horizontal plane can be interpreted by means of the dynamic considerations that have been presented. If a tongue of low temperature is present, as shown in fig. 126, one can assume in most cases that the water of the lowest temperature also has the highest density and, in accordance with the rule that in the Northern Hemisphere the water of the highest density must be found on the left-hand side of the current, arrows representing the approximate direction of flow can be entered. In the example shown in fig. 126 the flow to the northwest on the coastal side of the tongue of cold water was confirmed by results of drift-bottle experiments (Sverdrup and Fleming, 1941). Thus, the stream lines tend to follow the contour lines of the tongues.

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If the dynamics of the system were not borne in mind, one might assume that the spread of the water would be jetlike and that the axes of the jet currents would coincide with the tongues of low or high temperatures, as indicated by the open arrows in fig. 126. Such an interpretation, however, would neglect the effect of horizontal pressure gradients and therefore would be in disagreement with the character of the acting forces. These conclusions will be somewhat modified by a consideration of the effect of friction, and they may not be valid at the sea surface, where external factors may contribute toward maintaining the temperature distribution.

Consider next a vertical section in which a tonguelike distribution of temperature and salinity appears (fig. 127). In this case the tongue may indicate horizontal flow in the direction of the tongue, and such flow may no longer be in conflict with the dynamics of the system. However, if only horizontal motion occurs, processes of mixing must be of importance to the development of the tongue, because the properties of the mass in a moving volume are changing in the direction of flow, and in the subsurface layers such changes can be due only to mixing (p. 158).

Defant (1929b) and Thorade (1931) have examined analytically the relation between the velocity and the effect of vertical mixing. Under stationary conditions the equation (p. 159)

must be fulfilled if the distribution is maintained only by horizontal

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flow and vertical mixing. Here A represents the coefficient of eddy diffusivity, and V_x is the horizontal velocity.

Equation (XIII, 86) can be solved if A/ρ is supposed to be constant and a number of simple assumptions are introduced. Consider, for instance, a horizontal current of uniform velocity, ν, in a layer of thickness 2h, flowing between z = +h and z = − h (see fig. 127). Assume that at x = 0 the distribution of the property is given by the equation

and that at z = +h and z = −h—that is, at the upper and lower boundaries of the current—the property is constant, s = s₀. Provided that these conditions are fulfilled and that the character of the current is dynamically possible, a stationary distribution of the property can exist if

Figure 127 shows an example of such a distribution which has been prepared by Defant after the introduction of numerical values of s₀, Δs, and h and by assuming that A/ν = 2. The solution presupposes certain boundary conditions that are not specified here. Transport of the property into the volume must take place through the surface x = 0, and transport out of the volume must take place through the other boundary surfaces. In these circumstances, it is necessary to assume that certain physical processes maintain the constant value s = s₀ along the horizontal boundaries.

As shown by Thorade (1931), equation (XIII, 86) can be integrated on many different assumptions, and different types of tongues can be derived. It is particularly interesting to observe that on certain assumptions the tongue is found to curve up or down, although horizontal flow is assumed, and that the axis of the tongue need not coincide with an axis of maximum velocity. In fig. 127, uniform velocity was supposed, but the tonguelike distribution of the property may create the erroneous impression that the velocity is at a maximum along the center line of the tongue. In view of these results, it is necessary to exercise great care when drawing conclusions as to currents from tonguelike distribution.

Defant (1936a) has made use of equation (XIII, 86) in an improved form in order to compute the ratio A/ν directly from oceanographic data. Considering that along the bottom the motion cannot be horizontal, but must follow the slope of the bottom and that, even at some distance from the bottom, the motion may deviate from the horizontal, Defant writes

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where α is the angle that the stream lines form with the horizontal. Provided that the distribution of s is known, equation (XIII, 88) contains two unknowns: A/ρν and α. The continuity of the system, however, introduces restrictions as to the possible values that may fit the equation, and, by means of a series of trials, Defant could determine the probable direction of flow and the values of A/ρν in the deep water of the South Atlantic Ocean.

It has so far been assumed that horizontal mixing can be disregarded, but a tonguelike distribution can equally well be brought about by horizontal and vertical mixing and by horizontal flow and vertical mixing. If a given distribution in a vertical plane is maintained only by processes of mixing in horizontal and vertical directions, equation (XIII, 86) must be replaced by

provided that the coefficients of eddy diffusion, A_z and A_x, can be considered constant. Introducing the same assumptions that were made in order to present a solution of (XIII, 86), one obtains This solution is identical with (XIII, 87) if With the same numerical values as before, A/ν = 2 and h = 300 m = 3 × 10⁴ cm, one obtains The distribution of salinity shown in fig. 127 can therefore be brought about equally well by processes of horizontal mixing, provided that the horizontal eddy diffusivity is 9 × 10⁷ times greater than the vertical. Earlier studies gave no indication of the existence of such a tremendous horizontal eddy diffusion, but recent investigations (table 66 p. 485) indicate that within any current numerous large eddies are present which appear in the relative distribution of pressure, and that possibly converging or diverging winds may lead to piling up or removal of mass and thus induce irregular slope currents. Similarly, internal waves (see p. 601) may provide an effective stirring mechanism. Therefore the possibility exists that the horizontal eddy diffusion cannot be neglected, as had been previously supposed, but that the effect of the horizontal

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diffusion is of the same order of magnitude as the effect of the vertical. If this is true, no conclusions as to horizontal flow can be drawn from tonguelike distributions in vertical sections, unless the effect of mixing in both horizontal and vertical directions can be independently evaluated.

The “core method,” which Wüst has introduced (p. 146) and used successfully in studying the deep-water flow in the Atlantic, does not contain any assumption as to the character of the mixing, and is well suited therefore for giving a qualitative picture of the spreading of certain water types.