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METHODS FOR ESTIMATING DYNAMIC SLOPES AND CURRENTS IN SHALLOW WATER 1

PIER GROEN

Section of Oceanography Koninklijk N ederlands M eteorologi8ch lnstituut

In computing relative currents in the sea from the mass distribution, a well known difficulty is encountered when a section extends into rela-tively shallow water; then a mass of land becomes part of the section. To compute the relative currents or the relative slope of any isobar of the section with respect to, say, the 500 db isobar, the ordinary rules cannot be applied for points below which the pressure at the bottom is less than 500 db.

First Method. To overcome this difficulty , Helland-Hansen (1934) proposed a method which replaces the area of the cross section occupied by the solid earth by a fictitious section of water within which the lines of equal specific volume are horizontal (or isobaric2) continuations of the corresponding isosteric curves in the water.

Second Method. Sverdrup, Johnson and Fleming (1942: 451) pro-posed a different method, which they describe as follows: "As another approximation, one can apply the equation ip = - 'h (01 - 02) (p. 412), introducing the slope of the o surfaces where they cut the bottom and taking the difference o1 - o2 along the bottom" (italics mine). In the formula cited, ip is the relative slope of any isobar p = Pi with respect to a reference isobar p = P2, 01 and 02 designate the specific volume anomaly o at p = p1 and at p = p2 respectively, and i1 is the average inclina-tion of the o-lines between Pi and p2.

In a more general way we might describe the essence of this method as follows: In order to compute the relative slope of any isobar p = Pi with respect to the reference isobar p = P2 at a certain point P of the

1 Although the author had no discussions with Professor Sverdrup on the present matter during a visit of three and a half months at the Scripps Institution, it is a great pleasure to express sincere thanks for the stimulating contacts and to contribute this small note in his honor.

1 The difference between " horizontal" and "isobaric" is unimportant in this connection.

(313)

314 Journal of Marine Research [VII, 3

section, one uses the values of the specific volume a (or the anomaly 8) and the inclinations of the a-lines (or o-lines3

) along the vertical through P down to the bottom; beneath the bottom one projects horizontally onto this vertical the a-values (or the a-values) and the inclinations of the a-lines (or a-lines) occurring along the bottom-line in the cross section down to p = P2•

Now this procedure does not have the disadvantage of using a hypothetical physical picture of a rather arbitrary and somewhat un-likely sort (the barotropic, fictitious water-mass, which Helland-Hansen introduces). It has a disadvantage, however, in not corre-sponding to any physical picture at all, for it is impossible to replace the block of solid earth by a water mass such that the relative slopes of the isobars would everywhere be the same as the slopes computed by Sverdrup's method, except if the slopes of the isosteric lines should happen to be zero at all points of the bottom-line.

Indeed, in order to obtain a physical picture of this method of com-putation we would have to substitute for the block of solid earth a water mass which has along every vertical the horizontally projected values of the specific volume a and of the inclination of the a-lines occurring along the bottom; consequently, along any horizontal (or isobaric) line (any dotted line in l'ig. 1) in this fictitious water mass, a as well as the inclination of the a-lines would have to be constant. Such a water mass could exist physically only for the special case where the a-lines in the actual water meet the sloping bottom hori-zontally.

Proposed Third Method. Fixing the inclination of the a-lines at every point of the fictitious water mass alone determines a whole a-field (if only the values of a along the bottom-line are given). There-fore, we may propose, as a third method, the following procedure:

The block of solid earth is replaced by a fictitious section of water, in which the density distribution is determined by putting the inclina-tion of the a-lines along every horizontal (or isobaric) line equal to the value projected horizontally (or "isobarically") from the bottom-line. The a-lines are then extended from the actual water into the fictitious water by reference to these inclinations. The a-lines determined in this manner are shown as dashed lines in Fig. 1.

Instead of the specific volume we may also use its anomaly o; the

1 Of course, the a-line and the o-line have difierent slopes at any given point, which to use depends on the formula used.

1948] Groen: Estimating Dynamic Slopes and Currents 315

lines in Fig. 1 then represent a-lines. If, for instance, we use the approximate formula cited above,

ir, = - fa (01 - 02)1

then the difference of this method from Sverdrup's will be in the use of a different value of 021 namely a smaller one, if o in the actual water increases away from the bottom along the reference level, and a larger one in the opposite case.

---------~ -'"...,...;;--~- ---- - --~ ---------------- - --- - -- ------ -- _-_ \ ______ _

_ .., ___ _,.,,.,,.. ___ - --- -~- - -- ---~ _:~~

---

Figure I. Vertical cross section extending into shallow water. The scales are arbitrary• The fully drawn curves are a-lines; their continuations to the left of the bottom-line (dashed curves) have been drawn in such a way that at each level they have the sameslo:;;>e (Indicated by the straight little lines in the picture) as the a-line which meets the bottom-line at this level.

Discussion. Finding the value of the specific volume along a ver-tical in the fictitious water mass is not a matter in which high accuracy

316 Journal of Marine Research [VII, 3

must be aimed at, for this third procedure does not, of course, pretend to give the correct answer.

As a matter of fact, no method of solving the problem can prove its correctness, since, strictly speaking, the problem is not exactly defined. Since the question is to find the relative slope of a certain isobar at a certain point with respect to a reference isobar that does not exist vertically below this point, the only thing we may do is to draw up a procedure which is not too complicated, and which, if possible, has some physical background that is not too unlikely.

Finally, we should not think that, when knowing the relative slopes or currents at two different points at the same level with respect to one reference isobaric surface, we also know the relative slopes or currents at those points with respect to each other, unless we know, from other sources, the relative current distribution within the reference surface.

REFERENCES HELLAND-HANSEN, B1.,

1934. The Sognefjord section. James Johnstone Memorial Volume. Liverpool, Univ. press: 257-274.

SVERDRUP, H. u., M. w. JOHNSON AND R.H. FLEMING

1942. The oceans. Prentice-Hall, Inc., New York. 1087 pp.