As the amount of water flowing into a space must equal the amount of water flowing out of it per unit time, the rate of flow, i.e. velocity, is also important in continuity considerations. For example, a broad, shallow current entering narrow straits will become faster as well as perhaps becoming deeper.

QUESTION 4 9 1 \plain briefly how the flow pattern ol the subtropical gyres exemplifies the principle <il continuity.

The mathematical equation used to express the principle of continuity is:

du dv dw

dx dy

Figure 4.17 Schematic diagrams to illustrate one-and two-dimensional models for relatively simple, small-scale situations. Note that in these circumstances, the coordinate system shown in Figure 4.15 is applied differently, in that x and uare used for flow along the channel, and y and v are used for flow across the channel.

(a) A one-dimensional model for investigating flow through a channel. Such a model would require information about cross-sectional area (Ah A2, ...) and the average current speed (u1: u2, u3, ...) through each cross-section.

(b) A two-dimensional model of the channel can take account of cross-channel flow, vh v2, v3,..., between adjacent grid boxes, as well as along-

channel flow, uh u2, u3 which can vary across the channel. The model takes account of flow into and out of each side of each grid box (with the exception of those corresponding to fixed boundaries), but flow arrows are shown only for the nearest row of boxes. For clarity, flows vh v2, are showing at the front, but in reality they would be through the centres of the sides of the boxes. For a situation like an estuary, the grid boxes might have sides of-10-100 m.

which simply means that any change in the rate of flow in (say) the ^-direction must be compensated for by a change in the rate of flow in the y- and/or c-direction(s). This continuity equation is used in conjunction with the equations of motion and provides extra constraints, enabling the equations to be solved for whatever dynamic situation is being investigated.

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