WHY IS THERE A GULF STREAM?

We will now summarize briefly, with as little mathematics as possible, some of the developments in ideas that have led to an increased understanding of the dynamics of ocean circulation in general, and of the subtropical gyres in particular. In the course of this discussion, we will go a substantial way towards answering the question posed above: Why is there a Gulf Stream?

Ekman's initial work on wind-driven currents, intended to explain current flow at an angle to the wind direction, was published in 1905. In 1947, the Scandinavian oceanographer Harald Sverdrup used mathematics to demonstrate another surprising relationship between wind stress and ocean circulation. In constructing his theory, Ekman had assumed a hypothetical ocean which was not only infinitely wide and infinitely deep, but also had no horizontal pressure gradients because the sea-surface and all other isobaric surfaces were assumed to be horizontal. Sverdrup's aim was to determine current flow in response to wind stress a n d horizontal pressure gradients. Unlike Ekman, Sverdrup was not interested in determining how horizontal flow varied with depth; instead, he derived an equation for the total or net flow resulting from wind stress.

Consider a simple situation in which the winds blowing over a hypothetical Northern Hemisphere ocean are purely zonal and vary in strength with latitude sinusoidally, as shown in Figure 4.9(a).

*Zonal means either from east to west or from west to east; the equivalent word for north-south flow is 'meridional'.

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Figure 4.12 Summary of the results of Stommel's calculations. The diagrams on the left-hand side show the streamlines parallel to which water flows in the wind-driven layer (assumed to be 200 m deep). The volume of water transported around the gyre per second between one streamline and the next is 20% of the total flow. The diagrams on the right-hand side show contours of sea-surface height in cm. In (1), the ocean is assumed to be on a non-rotating Earth; in (2), the ocean is on a rotating Earth but the Coriolis parameter is assumed to be constant with latitude; in (3), the Coriolis parameter is assumed to vary linearly with latitude.

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This is how Stommel showed that the intensification of western

boundary currents is, in some way, the result of the fact that the Coriolis parameter ~'aries with latitude, increasing from the Equator to the poles.*

Stommel also explained intensification of western boundary currents in terms of vorticity balance. (In some ways, it is more convenient to work with vorticity than with linear current flow, because horizontal pressure gradient forces, which lead to complications, do not need to be

considered.)

Imagine a subtropical gyre in the Northern Hemisphere, acted upon by a symmetrical wind field like that in Figure 4.11. If the wind has been blowing for a long time, the ocean will have reached a state of equilibrium, tending to rotate neither faster nor slower with time: at every point in the ocean, the relative vorticity has a fixed value. This means that, over the ocean as a whole, those factors which act to change the relative vorticity of the moving water must cancel each other out.

Assuming for convenience that the depth of the flow is constant, we will now consider in turn each of the factors that affect relative vorticity.

The most obvious factor affecting the relative vorticity of the water in a gyre is the wind. The wind field is symmetrical and acts to supply negative (clockwise) vorticity over the whole region.

The next factor to consider is change in latitude. Water moving northwards on the western side of the gyre is moving into regions of larger positive planetary vorticity and hence acquires negative relative vorticity (cf. Figure 4.7(a)): similarly, water moving southwards on the eastern side of the gyre is moving into regions of smaller positive planetary vorticity and so loses negative relative vorticity (or gains positive relative vorticity). However, because as much water moves northwards as moves southwards, the net change in relative vorticity of water in the gyre as a result of change in latitude is zero.

The answer is friction. We may assume that the wind-driven circulation is not frictionally bound to the sea-floor, but there will be significant friction with the coastal boundaries as a result of turbulence in the form of (horizontal) eddies. Using appropriate values of Ah, the coefficient of eddy viscosity for horizontal motion (Section 3.1.1), we may make vorticity-balance calculations for a symmetrical ocean circulation. Such calculations show that for the frictional forces to be large enough to provide sufficient positive relative vorticity to balance the negative relative vorticity provided by wind stress, the gyral circulation would have to be many times faster than that observed in the real oceans.

* The variation of the Coriolis force with latitude is often referred to as the ~-effect.

Figure 4.13(a) is a pictorial representation of the vorticity balance of a symmetrical Northern Hemisphere gyre (cf. Figure 4.12(2)) that carries water around at the rate observed in the real oceans. In this gyre, there is an approximate vorticity balance in the eastern part of the ocean. The negative relative vorticity supplied to the water by the wind is nearly cancelled out by the positive relative vorticity that results from fi'iction with the eastern boundary combined with that gained by the water as a result of moving into lower latitudes. In the western part of the ocean, however, the combined effect of the negative relative vorticity supplied by the wind and that resulting from the movement of water into higher latitudes far outweighs the positive relative vorticity provided by friction. There is a continual gain of negative relative vorticity in the western part of the ocean and the gyre will accelerate indefinitely.

Figure 4.13(b) shows the corresponding diagram for a strongly asymmetrical current system in which water flows north in a narrow boundary current in the western part of the ocean and flows south over most of the rest of the ocean (cf. Figure 4.12(3)). In this asymmetrical gyre, the vorticity balance may be maintained in both the eastern and the western part of the ocean. In the eastern part of the ocean, where friction with the boundary is small, the vorticity balance is hardly affected at all, but on the western side, flow must now be much faster and the effects of both friction and change in latitude increase significantly. The negative relative vorticity resulting from change in latitude is acquired at a greater rate because water is now changing latitude much faster, and the gain of positive relative vorticity through friction is increased because both velocity and velocity shear have increased.* Both the planetary and frictional relative vorticity tendencies (which oppose one another) increase by an order of magnitude at least, and a vorticity balance is attained in the western ocean.

The result demonstrated in Question 4.7 and the vorticity-balance explanation given above are not mutually exclusive. Both are concerned with the effect on ocean circulation of the latitudinal variation in the angular velocity of the surface of the Earth. In the first case, this variation in angular velocity is represented by the variation of the Coriolis force with latitude, and in the second it is represented by the variation in planetary vorticity with latitude (cf. Figure 4.7(a)).

The theoretically derived pictures - or models - of ocean circulation derived by Sverdrup and Stommel (Figures 4.10 and 4.12(3)) bear a strong resemblance to the circulatory systems of the subtropical gyres. These circulation models were extended and made even more realistic by Walter Munk (1950). Like Stommel, Munk used a rotating rectangular ocean and assumed that the Coriolis force varied linearly with latitude, but he extended the ocean to include latitudes up to 60 ~ and the equatorial zone. He also balanced the relative vorticity supplied by wind, change of latitude and friction, but improved on the representation of both friction and the wind. As far as friction was concerned, he considered friction with coastal boundaries and friction associated with both lateral and vertical current shear, thus including the effect of eddy viscosity in both the horizontal and the vertical dimensions (i.e. both Ah and A" cf. Section 3.1.1).

*The fl'ictional force between moving water and a boundary is approximately proportional to the square of the cunent speed. This is analogous to the relationship between the frictional force of the wind on the sea-surface (the wind stress, "r) and the wind speed, W (Equation 3.1.1: = cVv "2) although, as mentioned later, in constructing the diagrams in Figure 4.12, Stommel used a simpler relationship, with friction directly proportional to current velocity.

SYMMETRICAL GYRE

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Figure 4.13 Pictorial representation of the various contributions to the vorticity of (a) a symmetrical subtropical gyre (cf. Figure 4.12(2)), and (b) a strongly asymmetrical subtropical gyre with an intensified western boundary current (cf. Figure 4.12(3)). The flow pattern in (b), in contrast to that in (a), enables a vorticity balance to be attained, so that the gyre does not rotate faster and faster indefinitely.

Figure 4.14 (a) The blue curve is the averaged annual zonal wind stress 1: for the Pacific and Atlantic Oceans; the black curve is the curl or torque of this wind stress, curl z. By definition, curl 1: is at a maximum at those latitudes where the wind stress curve shows the greatest change with latitude, e.g. at about 55~ where the wind stress changes from easterly to westerly, and at about 30 ~ N where it changes from westerly to easterly (cf. Figure 4.10(a)).

(b) The circulation pattern that Munk calculated using the values of curl "c. shown in (a). The volume transport between adjacent solid lines is 107 m 3 s -1. The greatest meridional flow occurs where curl 1: is at a maximum, i.e. at about 55 ~ 30 ~ and 10 ~ N; at these latitudes, flow is either southwards or northwards, rather than eastwards or westwards. (The 'West Wind Drift' is the old name for the Antarctic Circumpolar Current, which is of course, in the Southern Hemisphere -hence the quotation marks.)

Figure 4.14(b) shows the pattern of ocean circulation that Munk calculated.

If you compare this pattern with Figure 3.1 you will see that it bears a strong resemblance to the actual circulation observed in the North Pacific and the North Atlantic, so much so that it is possible to identify major circulatory features, as indicated on the right-hand side of the diagram.

Thus, in many ways, Munk's model of ocean circulation reproduces the circulation patterns seen in the real ocean. What this means is that the oceanic process that he and others considered important - namely flow of water into different latitudes (i.e. regions in which the Coriolis force/planetary vorticity is different), and horizontal and vertical frictional forces - really are important in determining the large-scale current patterns that result from the wind in the real oceans.

Models of ocean circulation are being improved and refined all the time.

Nevertheless, the basic 'tools' which oceanographers use to construct these models remain the same: they are the 'equations of motion'. The aim of Section 4.2.3 is to explain what these equations are and to convey something of how they are used. We have simplified the mathematical notation, but the general principles outlined form the basis for solving even very complicated dynamic problems.