Fx and Fy may include wind stress, friction or tidal forcing, depending upon the problem being investigated and the simplifications that can be made.
Note that the mathematical expressions used here have all come up already. The expressions for the horizontal pressure gradient forces are the same as those used in Chapter 3, and they have minus signs because the flow resulting from a horizontal pressure gradient is in the direction of decreasing pressure.
The expression for the Coriolis force has also been used already, although here mfu and mf\' have been replaced by p/i/ and p/i'. because we are considering the forces per unit volume.
Why does the equation for flow in the (-direction have p/i as the Coriolis term rather than pfu. and vice versa for flow in the y-direction?
The reason is. of course, that the Coriolis force acts at right angles to the current, so the component of the Coriolis force acting in the a-direction is proportional to the velocity in the y-direction and vice versa. (The minus sign before p/i/ is a consequence of the coordinate system in Figure 4.15; for example, in the Northern Hemisphere, the Coriolis force acting on water flowing in the positive .v-direction (i.e. towards the east) is in the negative v-direction (i.e. towards the south).)
As stressed already, the easiest situations to consider are those in which the ocean has reached an equilibrium or steady state, in which all the forces acting on the flowing water are in balance. In such situations, there is no acceleration and di//d? and dr/d/ in Equation 4.3 are zero. This means that the equations become very much easier to solve, especially if they can be simplified further.
The easiest way to simplify the equations is to assume that one or more of the forces concerned may be ignored altogether. For example, in his work on wind-driven currents Ekman assumed that the ocean was homogeneous and that the sea-surface was horizontal. This meant that there were no horizontal pressure gradients to worry about, and d/Vdv and dp/dy were zero.
Another way of keeping the equations simple is to express the contributions to F, and Fy simply. For example, we may decide to assume that friction is directly proportional to current speed, in which case it can be written as Au (for the .v-direction) and Bv (for the y-direction). where A and B are constants. This is the approach that Stommel adopted in calculating the flow patterns in Figure 4.12, which clearly showed that the intensificaton of western boundary currents can be explained by the variation of the Coriolis parameter with latitude.
The examples of the work done by Ekman and Stommel have been quoted 10 illustrate a particular point. It is this: breakthroughs in understanding of dynamic situations often come about as a result of someone realizing what lite most important variables) might be and then simplifying the situation under consideration (and hence the equations of motion) so that the effect of the particular aspect being investigated can be seen more clearly.
So far. we have not considered the vertical or .--direction. The main farces that must be considered here are the pressure gradient force in the vertical direction and. of course, the force resulting from gravity, written pf> rather than hi% because it is the weight per unit volume that we are interested in. The equation of motion for the --direction may therefore be written as:
Except beneath surface waves, vertical accelerations in the ocean arc generally very small, so for many purposes, dn/d/ may be assumed to be zero.
QUESTION 4.8 Rewrite Equation 4.3c assuming that du /d/ = 0, and that there are no forces acting vertically other than the weight of the water and tlie vertical pressure gradient force. Do you recognize the equation you have written?
Thus, when there are no vertical accelerations (other than jj), the equation of motion appropriate to the .--direction becomes simply the hydrostatic equation (Equation XB). indeed, tor large-scale motion, the hydrostatic equation is a very good approximation to the "vertical" equation of motion
At ibis point we should mention the principle of continuity, another important tool in dealing with mov ing water. The principle of continuity expresses the fact that mass must be conserved, i.e. the mass of water tin wing mio a given space must equal the mass of water (lowing out of that space. As seawater is virtually Incompressible, continuity of mass is effectively continuity of v olume, it follows that if the dimensions of a parcel of water change ill a particular direction (e.g. ihe v-direction in Figure 4 I fi). they must also change in one or both of the other directions. Thus, a wide shallow parcel of water will, on passing through narrow straits, become elongated and. if possible, deeper: alternatively, if a flow of water is obstructed, it may 'pile up", causing a local rise m sea-level.
figure i 16 Continuity ot volume during flow (here in the y-direc1ion>. Dimensions change but volume remains the same. Note that the parcel of water may change its shape as a result ot spatial constraints (e.g. bottom topography) or because of change in current speed: e.g. a parcel ot water emering a region of increased speed will become elongated in the direction of flow
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