## Uorticity

You will a I read) be familiar with the idea - essential (o an understanding of dynamic systems - thai energy and mass must be conserved. Another property that must be conserved is momentum both linear momentum associated with motion in straight lines and angular momentum associated with rotatory movement. In oceanography, it is more convenient to view the conservation of angular momentum as the conservation of a tendency to rotate, or (he conservation of vorlicity. ihe tendency to form vortices.

Ocean waters have rotatory motions on all scales, from the basin-wide subtropical and subpolar gyres down to the smallest swirls and eddies (e.g. Figures 4.4 and 3.31). Fluid motion does not have to be in closed loops to be rotatory: whenever there is current shear (a change in velocity at right angles to the direction of flow), there is a tendency to rotate and the water has vorticity. For mathematical convenience, a tendency to rotate anticlockwise is referred to as positive, and a tendency to rotate clockwise is referred to as negative. These aspects of vorticity are illustrated schematically in Figure 4.5.

Figure 4.4 Small-scale eddies forming in the vicinity of a rocky coastline.

Figure 4.4 Small-scale eddies forming in the vicinity of a rocky coastline.

NEGATIVE (CLOCKWISE) VORTICITY POSITIVE (ANTICLOCKWISE) VORTICITY

Figure 4.5 Diagrams to show examples of flow with (a) negative and (b) positive vorticity. The speed and direction of the flow are indicated by the lengths and directions of the arrows.

NEGATIVE (CLOCKWISE) VORTICITY POSITIVE (ANTICLOCKWISE) VORTICITY

Note that we say 'a tendency to rotate' rather than simply 'rotatory motion'. This is because water may be acquiring positive vorticity by one mechanism at the same time as it is acquiring negative vorticity by another. For example, water could be acquiring positive vorticity as a result of current shear caused by friction with adjacent bodies of water, or a coastal feature such as a headland or spit (Figure 4.6), while at the same time acquiring negative vorticity from a clockwise wind. The actual rotatory motion that results will depend on the relative sizes of the two effects. In theory, positive and negative vorticity tendencies could be exactly equal so that no rotatory motion would result.

Figure 4.5 Diagrams to show examples of flow with (a) negative and (b) positive vorticity. The speed and direction of the flow are indicated by the lengths and directions of the arrows.

anticlockwise eddy anticlockwise eddy

fricton with coastal boundary generates current shear and gives the (low positive voriicity direction of current (low

Figure 4.6 Diagram to illustrate how an eddy can be generated off a spit; the lengths and directions of the arrows indicate the speed and direction of the current. (The eddy on the left-hand side of Figure 4.4 is forming in the same way, though on a smaller scale.)

fricton with coastal boundary generates current shear and gives the (low positive voriicity direction of current (low

Figure 4.6 Diagram to illustrate how an eddy can be generated off a spit; the lengths and directions of the arrows indicate the speed and direction of the current. (The eddy on the left-hand side of Figure 4.4 is forming in the same way, though on a smaller scale.)

Water that has a rotatory motion in relation to the surface of the Earth, caused by wind stress and/or frictional forces, is said to possess relative vorticity. However, the Earth is itself rotating. The vorticity possessed by a parcel of fluid by reason of its being on the rotating Earth is known as its planetary vorticity.

### Planetary vorticity and the Coriolis force

In Chapter 1. you saw how the rotation of the Earth about its axis results in the deflection of currents and winds by the Coriolis force. These deflections were explained in terms of the poleward decrease in the eastward velocity of the surface of the Earth. Figure 1.2 showed how a missile fired northwards from the Equator is deflected eastwards in relation to the surface of the Earth because, with increasing latitude, the surface of the Earth travels eastwards at a progressively decreasing rate. This explanation is valid but it is only part of the story. In addition to a linear eastward velocity, the surface of the Earth also has an angular velocity, so that in the Northern Hemisphere it turns anticlockwise about a local vertical axis, and in the Southern Hemisphere it turns clockwise (Figure 4.7(a)). In other words, a cross marked on the Earth's surface, and viewed from a satellite in space positioned directly above it, would be seen to rotate.* Because the angular velocity of the Earth's surface is latitude-dependent, there would be relative motion between a hypothetical missile and the surface of the Earth, regardless of the direction in which the missile was fired (as long as it was not fired along the Equator), and this relative motion would increase with increasing latitude. Similarly, winds and currents moving eastwards and westwards experience deflection in relation to the Earth, as well as those moving northwards and southwards.

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