Note\ From now on, we will usually follow common practice and use the terms isobar and isopycnal - lines joining respectively points of equal pressure and points of equal density - instead of the more cumbersome 'isobaric surface' and 'isopycnic surface'.

You have seen that in geostrophic flow the slopes of the sea-surface, isobars and isopycnals are all related to the current velocity. This means that measurement of these slopes can be used to determine the current velocity.

In barotropic conditions, how might the geostrophic current velocity be estimated?

In theory, if the slope of the sea-surface could be measured, the current velocity could be determined using the gradient equation (Equation 3.11 ). In practice, it is only convenient to do this for flows through straits where the average sea-level on either side may be calculated using tide-gauge data - measuring the sea-surface slope in the open ocean would be extremely difficult, if not impossible, by traditional oceanographic methods.

In practice, it is the density distribution, as reflected by the slopes of the isopycnals, that is used to determine geostrophic current flow in the open oceans.

Why should it be relatively straightforward to determine ihe density distribution in a volume of ocean?

Because density is a function of temperature and salinity, both of which are routinely and fairly easily measured with the necessary precision. Density is also a function of pressure or. to a first approximation, depth, which is also fairly easy to measure precisely.

Measurement of the density distribution also has the advantage - as observed earlier - that in baroclinic conditions the slopes of the isopycnals are several hundred times those of the sea-surface and other isobars. However, it is important not to lose sight of the fact that it is the slopes of the isobars that we are ultimately interested in, because they control the horizontal pressure gradient; determination of the slopes of the isopycnals is, in a sense, only a means to an end.

Figure 3.16(a) and (b) are two-dimensional versions of Figure 3.15(a) and (b). In the situation shown in (a), conditions are barotropic and w, the current velocity at right angles to the cross-section, is constant with depth.

In these circumstances, how may we determine m?

We know that for any isobaric surface making an angle 0 with the horizontal, the velocity u along that surface may be calculated using the gradient equation:

g where /is the Coriolis parameter and g is the acceleration due to gravity.

As conditions here are barotropic, the isopycnals are parallel to the isobars and we can therefore determine 0 by measuring the slope of the isopycnals (or the slope of the sea-surface). The velocity, u, will then be given by:

at all depths in the water column. The depth-invariant currents that flow in barotropic conditions (isobars parallel to the sea-surface slope) are sometimes described as 'slope currents'; they are often too small to be measured directly.

By contrast, the geostrophic currents that flow in baroclinic conditions vary with depth (Figure 3.15(b)). Unfortunately, from the density distribution alone we can only deduce relative current velocities; that is, we can only deduce differences in current velocity between one depth and another. However, if we know the isobaric slope or the current velocity at some depth, we may use the density distribution to calculate how much greater (or less) the isobaric slope, and hence geostrophic current velocity, will be at other depths. For convenience, it is often assumed that at some fairly deep level the isobars are horizontal (i.e. the horizontal pressure gradient force is zero) and the geostrophic velocity is therefore also zero. Relative current velocities calculated with respect to this level - known as the 'reference level' or 'level of no motion' - may be assumed to be absolute velocities. This is the approach we will take here.

Now look at Figure 3.16(b), which we will again assume represents a cross-section of ocean at right angles to the geostrophic current. A and B are two oceanographic stations a distance L apart. At each station, measurements of temperature and salinity have been made at various depths, and used to deduce how density varies with depth. However, if we are to find out what the geostrophic velocity is at depth z\ (say), we really need to know how pressure varies with depth at each station, so that we can calculate the slope of the isobars at depth z\.

Figure 3.16 (a) In barotropic conditions, the slope of the isobars is tan e at all depths; the geostrophic current velocity u is therefore (git) tan 9 (Eqn 3.11 a) at all depths, (b) In baroclinic conditions, the slope of the isobars varies with depth. At depth z\ the isobar corresponding to pressure pi has a slope of tan 61. At depth z0 (the reference level), the isobar corresponding to pressure p0 is assumed to be horizontal. (For further details, see text.)

Figure 3.16 (a) In barotropic conditions, the slope of the isobars is tan e at all depths; the geostrophic current velocity u is therefore (git) tan 9 (Eqn 3.11 a) at all depths, (b) In baroclinic conditions, the slope of the isobars varies with depth. At depth z\ the isobar corresponding to pressure pi has a slope of tan 61. At depth z0 (the reference level), the isobar corresponding to pressure p0 is assumed to be horizontal. (For further details, see text.)

Wh\ do we want to know the slope of the isobars?

So that we can apply the gradient equation (3.11a), and hence obtain a value for u.

Assume that in this theoretical region of ocean the reference level has been chosen to be at depth zo- From our measurements of temperature and salinity, we know that the average density p of a column of water between depths zi and zo is greater at station A than at station B; i.e. pA > Pb- The distance between the isobars p\ and po must therefore be greater at B than at A, because hydrostatic pressure is given by pgh, where h is the height of the column of water (cf. Equation 3.8). Isobar p\ must therefore slope up from A to B, making an angle 0i with the horizontal, as shown in Figure 3.16(b).

How can tan รถ| be expressed in terms of dislances shown on Figure 3.16(b)?

It is given by tan 0

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