## Question

(a) Hydrographic stations A and B are 100 km apart at 30° of latitude. Temperature and salinity measurements at the two stations indicate that pA= 1.0265 x lO'kgnr3 and pa = l.0262x lO'kgm \ Calculate the geostrophic velocity u ai depth n = 1000m, biking your reference level (rni as 2000 m (in order to apply Equation 3.13 you will have to make the assumption that hs and :u - are equal). The value for g is M.K m s

(hi (i) Use Equation 3,12 to calculate the difference h\i - li\. How valid was the assumption, made above, lhal lih = ;o - ."i? (ii) Use your value for - lt \ to calculate tan H (i.e. the slope of the isobar at 1000 m depth), and the angle 0 itself.

(c) II stations A and B are in the Southern Hemisphere, with B due east of A. in which direction is the geostrophic current flowing? Given that the conventional symbols for indicating How direction are ® for 'flow into the page' and 0 for 'flow out of the page', draw a simple diagram to illustrate this situation (A on the left. B on the right). Show the sloping sea-surface and the sloping isobar at 1000m depth, and arrows representing the horizontal pressure gradient force and the Coriolis force either side of ihe appropriate symbol for How direction.

Note: Before moving on, \<>it should ensure that you ¡.'¿in follow the answer to Question 3.7. especially putt (cl The sketch for this is shown overleaf, in Figure 3. I7ta).

By selling Iih equal to - in Question 3.7|a). you were effectively using average densities p \ and pu ihat had been calculated assuming that the columns of water between isobars p\ and p\i were the same height at A and B. As you will have found in part (b). the difference in height between the two columns C' 11 ~ ha) is a very small fraction (0.03 9r) of :n and the resulting error in Ph. and hence in ¡r. is also extremely small.

In Question 3.7fu I you calculated the velocity of the geostrophic current at one depth. .-¡.A complete profile of the variation of the geostrophic current velocity with depth may be obtained by choosing a reference level and then applying Equation 3.13 at successively higher levels, each tunc calculating the geostrophic current velocity in relation to the level below (see for example Figure 3.18(a)). This is how geostrophic current velocities have been calculated since the beginning of the 20th century: the full version of Equation 3.13 used for this purpose is often referred to as He 11 and-Hansen's equation, after the Scandinavian oceanographer Bjorn He Hand-Hansen who. with Johan W. Sandstrom, did much pioneering work in this field.

Until relatively recently, oceanographers had to calculate seawater densities from measured values of temperature, salinity and depth, using standard tables. Today, temperature, salinity and depth are routinely recorded electronically, and the necessary calculations (effectively, many iterations of the gradient equation) are done by computer, but ii is still important for oceanographers to be aware of the limitations of the geostrophic method and the Various assumptions and approximations that are implicit in the calculations

The first poim to note is that we have been implicitly assuming thai the geostrophic current we are attempting to quantify is flowing en right tingles to the section A-B. In reality, there is no way of ensuring that this is the case. 11 Ihe current direction makes an angle with the section, the geostrophic equation (3.13) will only provide a value for Ihat component of the How at right angles to it. The calculated geostrophic velocities will therefore be underestimates. To take an extreme example, if the average flow in a region is north-eastwards and the section taken is from the south-west to the north-east (i.e. effectively parallel to the flow ), the geostrophic velocity calculated will be only a very small proportion of the actual geostrophic velocity to the north-east. To get over this problem» a second section may be made at right angles to the first, and the total current calculated by combining the two components. In practice, the geostrophic flow in an area is usually determined using data from a grid of stations.

The second important limitation of this method is that, as discussed earlier, it can only he used to determine relative velocity Thus in calculating the geostrophic current in Question 3.7(a). it was assumed ihat at a dcplh of 2000111 - the reference level - the current was zero. II in fact current measurements revealed that there was a current of. say. 0,05 m s1 at this depth, that value could be added on to the (calculated) velocity at all depths.

Current that persists below the chosen reference level is sometimes referred to as the barotropic' pan of the flow (Figure 3.18(b)). As mentioned in Section 3.3.2. in the deep ocean below the permanent ihermocline. density and pressure usually vary as a function of depth only, and so even if isobars and isopycnals are not horizontal, they are likely to he parallel to one another.

Would the barotropic pad of the flow show up ¡11 calculations like those you made in (Question 3.7(a)?

Co rio lis force

Co rio lis force

Figure 3.17 (a) Answer to Question 3.7(c). Note that the pressure gradient arrows are horizontal. Remember that the sketch relates to the Southern Hemisphere. (b) Drawing to illustrate the rationale for the conventional current-direction symbols, ® and O.

No. The geostrophic velocity u obtained using Equation 3.13 is that for flow resulting from lateral variations in density (i.e. attributable to the difference between pA and ps). The effect of any horizontal pressure gradient that remains constant with depth is not included in Equation 3.13. In reality, 'barotropic flow' resulting from a sea-surface slope caused by wind, and 'baroclinic flow' associated with lateral variations in density - or, put another way. the 'slope current' and the 'relative current' - may not be as easily separated from one another as Figure 3.18 suggests.

Another point that must be borne in mind is that the geostrophic equation only provides information about the average flow between stations (which may be many tens of kilometres apart) and gives no information about details of the flow. However, this is not a problem if the investigator is interested only in the large-scale mean conditions. Indeed, in some ways it may even be an advantage because it means that the effects of small-scale fluctuations are averaged out, along with variations in the flow that take place during the time the measurements are being made (which may be from a few days to a few weeks).

We have seen how information about the distribution of density with depth may be used to determine a detailed profile of geostrophic current velocity with depth. Although both density and current velocity generally vary continuously with depth (e.g. Figure 3.19(a) and Figure 3.18), for some purposes it is convenient to think of the ocean as a number of homogeneous layers, each with a constant density and velocity. This simplification is most often applied in considerations of the motion of the mixed surface layer, which may be assumed to be a homogeneous layer separated from the deeper, colder waters by an abrupt density discontinuity (Figure 3.19(b)), rather than by a pycnocline - an increase in density over a finite depth. In this situation, the slopes of the sea-surface and the interface will be as

Figure 3.18 (a) Example of a profile of geostrophic current velocity, calculated on the assumption that the horizontal pressure gradient, and hence the geostrophic current velocity, are zero at 1000 m depth (the reference level).

(b) If direct current measurements reveal that the current velocity below about 1000 m is not zero as assumed but some finite value (say 0.05 m s 1), the geostrophic current velocity profile would look like this. The geostrophic velocity at any depth may therefore be regarded as a combination of baroclinic and barotropic components.

Figure 3.18 (a) Example of a profile of geostrophic current velocity, calculated on the assumption that the horizontal pressure gradient, and hence the geostrophic current velocity, are zero at 1000 m depth (the reference level).

(b) If direct current measurements reveal that the current velocity below about 1000 m is not zero as assumed but some finite value (say 0.05 m s 1), the geostrophic current velocity profile would look like this. The geostrophic velocity at any depth may therefore be regarded as a combination of baroclinic and barotropic components.

baroclinic vetoeity from geostrophic calculation velocity (m 5" 0.2 I

'barotropic velocity'

which does not show up in calculation velocity (ms"1) 0 0.1 0.2 0.3

baroclinic vetoeity from geostrophic calculation

'barotropic velocity'

which does not show up in calculation velocity (m 5" 0.2 I

shown in Figure 3.20(a) and, for convenience, the geostrophic velocity of the upper layer may be calculated on the assumption that the lower layer is motionless. Figure 3.20(b) is an example of a more complex model, which may also sometimes approximate to reality; here there are three homogeneous layers, with the intermediate layer flowing in the opposite direction to the other two.

In Chapter 5, you will see how such simplifications may help us to interpret the density and temperature structure of the upper ocean in terms of geostrophic current velocity.

density

Figure 3.19 (a) Typical density profiles for different latitudes: solid line = tropical latitudes; dashed line = equatorial latitudes; dotted line = high latitudes.

(b) The type of simplified density distribution sometimes assumed in order to estimate geostrophic currents in the mixed surface layer.