In answering Question 4.6(b) and (c), you considered two types of current flow resulting from wind stress:

(i) Ekman transport at right angles to the direction of the wind, and

(ii) geostrophic flow, in response to horizontal pressure gradients caused by the 'piling up' of water through Ekman transport.

Sverdrup combined these components mathematically and obtained the flow pattern shown in Figure 4.10(b). The 'gyre' is not complete because in deducing this flow pattern, Sverdrup considered the effect of an eastern boundary but could not also include the effect of a western boundary. As a result, there could be no northward flow in a 'Gulf Stream'.

The most interesting aspect of Sverdrup's results, however, was that the net amount of water transported by a given pattern of wind stress depends not on the absolute value of the wind stress but on its torque, i.e. its tendency to cause rotation - or, if you like, its ability to supply relative vorticity to the ocean. In particular, Sverdrup showed that the net amount of water transported meridionally (i.e. north-south or south-north) is directly proportional to the torque of the wind stress. You can see this for yourself, in a qualitative way, by studying Figure 4.10. The first thing to note is that the pattern of wind stress shown in part (a) of Figures 4.9 and 4.10 will result in a clockwise torque on the ocean (cf. Question 4.6(a)). This torque is greatest at latitude 30°, and it is here that the southward motion, as shown by the flow lines on Figure 4.10(b), is greatest. The torque is least (effectively zero) at latitudes 45° and 15°, and here the flow is almost entirely zonal (eastwards and westwards, respectively).

(b) The depth-integrated' {i.e net. when summed over the depth) flow patter« that results from combining Ekman transports at right angles to the wind with geostrophii; (low in response to horizontal pressure gradient farces, according to Sverdrup

(b) The depth-integrated' {i.e net. when summed over the depth) flow patter« that results from combining Ekman transports at right angles to the wind with geostrophii; (low in response to horizontal pressure gradient farces, according to Sverdrup

In summary then- Sverdrup showed lhal at any location in the ocean, I he total amount of water transported meridionally (n the wind-influenced layer is proportional to ihc lot que of the w ind stress. If the meridional transport is given the symbol M. this can be wrillen as:

M-bx torijueof X t-l.la)

where r is the w ind stress, b is a latitude-dependent constant and the utttts of M are kg s-1 per m. or kg m"1 s The word used in mathematics for "torque' is 'curl', so Equal ion 4. la can be written as:

Sverdrup found that the constant in Equation 4.1 is the reciprocal of the rate of change of the Coriolis parameter with latitude. The rate of change of/with latitude is commonly given the symbol |i, so Equation 4.1 is usually written as:

You are not expected lo manipulate this equation, but you should appreciate that it is a convenient way of ¡summarizing Sverdrup's theoretically determined relationship, namely that at any location in the ocean the total meridional flow - i.e. ihe net north-south flow, taking into account the speed and direction of flow at all depths, from the surface down to the depth ai which even indirect effects of'lhe wind cannot be felt - is determined by the rale of change off'at the latitude concerned and the torque, or curl, of the wind stress.

Sverdrup's theory was an advance on Ekman's in that the ocean was not assumed to be unlimited. However, as mentioned above, it could not take account of a western boundary, and certainly could r>ol explain the existence of intense western boundary currents like the Gult Stream (Figure 4.11}. This problem was solved by the American oceanographer Henry Stomincl, in a paper published in lL)4K. Stommel considered the effect of a symmetrical anticyclonic wind Held on a rectangular ocean in three different situations:

1 The ocean is assumed lo be on a non-rotating Harth

2 The ocean is rotating but the Cork) I is parameter./ is constant.

3 The ocean is rotating and the Coriplis parameter vanes with latitude (far the sake of simplicity, this variation is assumed to be linear).

Unlike Sverdrup. Sloiiimel included friction in his calculations and he worked out the flow that would result when the wind stress and frictional forces balanced (i.e. when die re was a steady stale, with no acceleration or deceleration! for each of the ihree situations described above.

Figure 4.12 (overleaf) summarizes the results of Slonimcrs calculations. The left-hand diagrams represent the patterns of flow that result from a symmetrical an I icy clonic wind Held in situations (I )-i3). and in each case the right-hand diagram shows the sea-surface topography that accompanies this flow

Look lirsi on the diagrams on the right-hand side, which show the sea-surface topography. What topographic feature may be seen clearly in situation (2). but is not present in situation (I)?

In (2). the sea-surface has a large 1150 cm) topographic high in llie middle of the gyre, whereas there is no such feature in (11. The reason lor the difference is that each diagram represents a situation of equilibrium, where forces balance, in (2). unlike (I). the ocean is on a rotating Larth and a Coriolis force has come into existence. For there lo be equilibrium. Ihis must be balanced by horizontal pressure gradient forces, w hich are brought about by the sea-surface slopes.

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