3.2 INERTIA CURRENTS
3.3.3 DETERMINATION OF GEOSTROPHIC CURRENT VELOCITIES
53 Barotropic conditions may be found in the well-mixed surface layers of the ocean, and in shallow shelf seas, particularly where shelf waters are well- mixed by tidal currents. They also characterize the deep ocean, below the permanent thermocline, where density and pressure are generally only a function of depth, so that isopycnic surfaces and isobaric surfaces are parallel.
Conditions are most strongly baroclinic (i.e. the angle between isobaric and isopycnic surfaces is greatest) in regions of fast surface current flow.
Currents are described as geostrophic when the Coriolis force acting on the moving water is balanced by the horizontal pressure gradient force. This is true whether the water movement is being maintained by wind stress and the waters of the upper ocean have 'rearranged themselves' so that the density distribution is such that geostrophic equilibrium is attained, o r whether the density distribution is itself the c a u s e of water movement.
Indeed, it is often impossible to determine whether a horizontal pressure gradient within the ocean is the cause or the result of current flow, and in many cases it may not be appropriate to try to make this distinction.
For there to be exact geostrophic equilibrium, the flow should be steady and the pressure gradient and the Coriolis force should be the only forces acting on the water, other than the attraction due to gravity. In the real oceans, other influences may be important; for example, there may be friction with nearby coastal boundaries or adjacent currents, or with the sea-floor. In addition, there may be local accelerations and fluctuations, both vertical (resulting perhaps from internal waves) and horizontal (as when flow paths are curved). Nevertheless, within many ocean currents, including all the major surface current s y s t e m s - e.g. the Gulf Stream, the Antarctic Circumpolar Current and the equatorial currents - flow is, to a first approximation, in geostrophic equilibrium.
It is important to remember that the slopes shown in diagrams like
Figures 3.10 to 3.15 are greatly exaggerated. Sea-surface slopes associated with geostrophic currents are broad, shallow, topographic irregularities.
They may be caused by prevailing winds 'piling up' water against a coastal boundary, by variations in pressure in the overlying atmosphere, or by lateral variations in water density resulting from differing temperature and salinity characteristics (in which case, conditions are baroclinic), or by some combination of these factors. The slopes have gradients of about 1 in
105 to 1 in l0 s, i.e. a few metres in 102-105 km, so they are extremely difficult to detect, let alone measure. However, under baroclinic conditions the isopycnic surfaces may have slopes that are several hundred times greater than this, and these c a n be determined. How this is done is outlined in Section 3.3.3.
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
o c e a n s .
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 e a r l i e r - 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 u, the current velocity at right angles to the cross-section, is constant with depth.
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:
tan 0 - fu (Eqn 3.11 )
g
w h e r e f i s 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:
u - g tan 0 (3.1 la)
f
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.
55 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 dens#3, varies with depth. However, if we are to find out what the geostrophic velocity is at depth zl (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 0 at all depths; the geostrophic current velocity u is therefore
(g/f) tan 0 (Eqn 3.11a) at all depths.
(b) In baroclinic conditions, the slope of the isobars varies with depth. At depth zl the isobar corresponding to pressure pl has a slope of tan el. At depth Zo (the reference level), the isobar corresponding to pressure Po is assumed to be horizontal. (For further details, see text.)
So that we can apply the gradient equation (3.1 la), 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 z0. From our measurements of temperature and salinity, we know that the average density P of a column of water between depths z~ and z0 is greater at station A than at station B; i.e. [3A > DB- The distance between the isobars pl and p0 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 Pl must therefore slope up from A to B, making an angle 01 with the horizontal, as shown in Figure 3.16(b).
9 "~~" .-. ~'~ ~ c~ V~'~< II - ~. .~ ~
~~_~.o ~1~
9 ~" m,,. ~. 9II ~-. II II ~
F ~ b~
h,,~. 9 o~
~.~
~ ~ ~ ~.~~ ~~ ~.~ ~ ~~
~Y~. 9~~~
c~~-~ ~.~ ~~-~
~ ~" ~~~o~~. ~~~:
---.-~ r~ ~'~~~ ~~ ~~ ~ ~. ~ "
oo~ ~ ~ ~. ~'~ ~ ~ ~ :~ ~. ~-~ ~ ~ ~ =.~ ~- <o~~ (~ ~..~. .~ ~ ~- -. ~ 9~.~ ~~
~ ~=~~'~
~~ ~.~ . ~- ~-=~ --- 9 ~ r~"~. ... ~ :::~" ~ ,,.~ ~ ~ ~ e~ ~~ ~ ~ o .~ ~ -~o ._~
=. .~ ~ ~ ~" ~-~" , ~~
~ ~.~ ~ ~ ~~~- ~. ~ ~ ~ ~ = = -= o ~~~c~~-~ ~ ~-y~~ ~ c~m ~c~ ~'~* ~ ~ ,~ ~ ~ ~ ~ ~_~- ~. ~.o ~ ~~ ~~ 9 ~ ~ ~ ,-~ :~~~ ~ ~~~. ~ ~ ~~.~~ o~ _.o
~-~ ~ ~_~.~. o~-~: ~ ~.~ ~ o ~~
~ ~ ~, ,-~ 0 ""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 |
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 PB). 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 - o r , put another way, the 'slope current' and the 'relative c u r r e n t ' - 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-l), 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.
59 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.
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.
1000 -
E .= 2000--
3000 --
40OO -- (a)
1.024 density
, ,
density (103 kg m -3)
1.026 1.028 ...
-- -- -.~ .~. ', I
\ \
r
(b)
Figure 3.20 Diagrams to show how the interfaces between layers slope in the case of (a) two and (b) three homogeneous layers, where pl < 92 < 93. The symbols for flow direction are drawn on the assumption that the locations are in the Northern Hemisphere.
s 9 --., 9 9 0 0 ~'~ N.,,. 0 "* 0 0 ~q O0 ,,-, ,-.]# ;:3" :::r~ 0 ==o
~'s 0 ~ ,--- ,-I -.,. . 9
~~~~-
,~ = ::::r' ~ o s~ ~~ 2. -~ ~.=~ ~ ~" ~ r~ ,-, ~ '--' 9 ,.~D 0 @ "-' ~ I;..~ ~- #. =-~ ~.~-~= ,~'~ ~= r~ .~ ~" 0 '' ''~ ~ ~ ["~'1 = '~ ~ =" =" = ;.= ~ 5.~ ._ 0 q-~= ,_~ ~.~ " o '~ ~'~ 5 ~
~~~
~ S ~ =-~'~ S" o =. = ~ ,.< ==o ~ ~." '~,.-~ ,~. ~" ~= = ~'--'~ =" ~.o o ~ ~ ~-" ,,-. ,-~ ,--,. , ~ =-~= -=o~-o-. r~ r~ 6"=. ~';.~= ~ ~ ~-~- 2~= = ,J-~
~ ~ !'11"1 ,, ,,~ ,~. r'r'l ;.= x ~ ~ ;::3" ~ ~" ~, ~ ~. ~ ..., ,'-'* 0 - "< ~. - o~ - ='=--, 5 F~
61 effects of variations in atmospheric pressure cannot be completely ignored, in theory they could easily be corrected for. However, weather systems may travel several hundred kilometres per day, so differences in atmospheric pressure between two stations are likely to fluctuate over the period in which measurements are made. In practice, therefore, the effect of atmospheric pressure on geostrophic current flow is difficult to take account of, and in most situations is considered sufficiently small to be ignored.
We deduced above that l0 m of seawater is equivalent to a pressure of 1 bar, i.e. that 1 m of seawater is equivalent to 1 decibar (dbar). For many purposes this is a very useful approximation. However, like all fluids, seawater is compressible, and for greater depths quite significant errors result from converting pressures in decibars directly to depths in metres. For example, a pressure of 11 240 dbar has been recorded at the bottom of the Marianas Trench, but soundings of the area indicate that the maximum depth is about 10 880 m.