THE EQUATIONS OF MOTION

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.

99 The e q u a t i o n s of m o t i o n - the equations that physical oceanographers need to solve in order to be able to describe the dynamics of the ocean - are simply N e w t o n ' s Second Law of Motion"

force - mass • acceleration (4.2a)

applied to a fluid moving over the surface of the Earth.

Equation 4.2a can be rearranged to give"

force acceleration -

mass

(4.2b)

In dealing with moving fluids, it is convenient to consider the forces acting 'per unit volume'. The mass of unit volume of fluid is given numerically by its density, 9 9 We can therefore write Equation 4.2b as"

acceleration - force per unit volume density

= - x force per unit volume 1 9

(4.2c~

In order to determine the characteristics of flow in three dimensions, we must apply Equation 4.2c in three directions at right angles to one another.

The x, v, " coordinate system used by convention in oceanography is shown in Figure 4.15. The velocities in the x-, y- and 7.-directions are u, v and w, respectively, and so the corresponding accelerations (rates of change of velocity with time) are du/dt, dv/dt and dw/dt.

Figure 4.15 (a) The coordinate system commonly used in oceanography. The x-axis is positive eastwards and negative westwards;

the y-axis is positive northwards and negative southwards; and the z-axis is positive upwards and negative downwards.

(b) The components of the current velocity in the x-, y- and -z-directions are, by convention, u, v and -w, respectively.

We hope you came up with the Coriolis force and the horizontal pressure gradient force, as well as perhaps wind stress and other frictional forces.

The equation of motion appropriate to flow in the x-direction (i.e. easterly or westerly flow) may therefore be written:

d., ((h~176 pressure) ( C~176 ~ /

= - gradient force in + resulting in motion dt P the x - direction in the x - direction

other forces ~,'~

related to motion I/

in the x - direction })

For flow in the v-direction (i.e. northwards or southwards flow), the left-hand side of the equation is d~'/dt and the right-hand side is identical to that above, except that 'in the x-direction' is replaced in each case by 'in the v-direction'.

In mathematical terms, the equations of motion for flow in the x- and v-directions may therefore be written:

pressure Coriolis contributions gradient force force from other forces

acceleration + { +

d . 1 ( 6 1 , )

- = - + pfi, + F, (4.3a)

dt p dx

dv dt - - l P ( d/, d,' - 9.h' + F , ) (4.3b)

F, and F,. may include wind stress, friction or tidal forcing, depending upon the probiem 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 raft, have been replaced by 9fu and 9,/i,. because we are considering the forces per unit volume.

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 x-direction is proportional to the velocity in the v-direction and vice ~'ersa. (The minus sign before 9fu 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 x-direction (i.e. towards the east) is in the negative y-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 du/dt and d~,/dt 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 dp/ch and dp/dy were zero.

Another way of keeping the equations simple is to express the contributions to F,. and F,. 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 x-direction) and B~, (for the v-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.

~oo~ ~

=~.ea~~

~o~~_~ -~

~ ~'~ ~.~ =~ ~> 2=2 ~

~.~

+ 9 c) .<.. ~_.. -- 9 m ,_,. C~ b~ C~ 9 t') 9 o

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.

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 u are 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 (A1, A2 . . . . ) and the average current speed (~1, ~2, ~3 . . . . ) through each cross-section.

(b) A two-dimensional model of the channel can take account of cross-channel flow, vl, v2, v3 ...

between adjacent grid boxes, as well as along- channel flow, ul, 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 v~, v2, v3 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.

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

du dv dw

+ ~ + ~ - 0

dx dy dz (4.3d)

which simply means that any change in the rate of flow in (say) the x-direction must be compensated for by a change in the rate of flow in the y- and/or ,z-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.

4.2.4 INVESTIGATING THE OCEAN THROUGH COMPUTER MODELLING