BAROTROPIC AND BAROCLINIC CONDITIONS

In the idealized situation discussed in the previous Section, a sea-surface slope had been set up across an ocean consisting of seawater of constant density. Consequently, the resulting horizontal pressure gradient was simply a function of the degree of slope of the surface (tan 6) - the greater the sea- surface slope, the greater the horizontal pressure gradient. In such a situation, the surfaces of equal pressure within the ocean - the isobaric surfaces - are parallel to the sea-surface, itself the topmost isobaric surface since the pressure acting all over it is that of the atmosphere.

In real situations where ocean waters are well-mixed and therefore fairly homogeneous, density nevertheless increases with depth because of compression caused by the weight of overlying water. In these circumstances, the isobaric surfaces are parallel not only to the sea-surface but also to the surfaces of constant density or isopyenie surfaces. Such conditions are described as b a r o t r o p i c (see Figure 3.11 (a)).

BAROCLINIC CONDITIONS

Figure 3.11 The relationship between isobaric and isopycnic surfaces in (a) barotropic conditions and (b) baroclinic conditions. In barotropic conditions, the density distribution (indicated by the intensity of blue shading) does not influence the shape of isobaric surfaces, which follow the slope of the sea-surface at all depths. By contrast, in baroclinic conditions, lateral variations in density do affect the shape of isobaric surfaces, so that with increasing depth they follow the sea-surface less and less.

pressure

gradient

force

m l l l l l l ~ ~ Coriolis force

\

\

\

\

Coriolis force

. ]

initial

~ situation pressure

gradient force

(b) PLAN

Figure 3.12 (a) A sea-surface slope up towards the east results in a horizontal pressure gradient force towards the west.

(b) Initially, this causes motion 'down the pressure gradient', but because the Coriolis force acts at right angles to the direction of motion (to the right, if this is in the Northern Hemisphere), the equilibrium situation is one in which the direction of flow (dashed blue line) is at right angles to the pressure gradient.

As discussed in Section 3.3.1, the hydrostatic pressure at any given depth in the ocean is determined by the weight of overlying seawater. In barotropic conditions, the variation of pressure over a horizontal surface at depth is determined only by the slope of the sea-surface, which is why isobaric surfaces are parallel to the sea-surface. However, any variations in the density of seawater will also affect the weight of overlying seawater, and hence the pressure, acting on a horizontal surface at depth. Therefore, in situations where there are lateral variations in density, isobaric surfaces follow the sea-surface less and less with increasing depth. They intersect isopycnic surfaces and the two slope in opposite directions (see Figure 3.11 (b)).

Because isobaric and isopycnic surfaces are inclined with respect to one another, such conditions are known as baroclinic.

Geostrophic c u r r e n t s - currents in which the horizontal pressure gradient force is balanced by the Coriolis f o r c e - may occur whether conditions in the ocean are barotropic (homogeneous), or baroclinic (with lateral variations in density). At the end of the previous Section we noted that in the hypothetical ocean where the pressure gradient force was the only horizontal force acting, motion would occur in the direction of the pressure gradient. In the real ocean, as soon as water begins to move in the direction of a horizontal pressure gradient, it becomes subject to the Coriolis force.

Imagine, for example, a region in a Northern Hemisphere ocean, in which the sea-surface slopes up towards the east, so that there is a horizontal pressure gradient force acting from east to west (Figure 3.12(a)). Water moving westwards under the influence of the horizontal pressure gradient force immediately begins to be deflected towards the north by the Coriolis force; eventually an equilibrium situation may be attained in which the water flows northwards, the Coriolis force acting on it is towards the east, and is balancing the horizontal pressure gradient force towards the west (Figure 3.12(b)). Thus, in a geostrophic current, instead of moving down the horizontal pressure gradient, water moves at right angles to it.

Moving fluids tend towards situations of equilibrium, and so flow in the ocean is often geostrophic or nearly so. As discussed in Section 2.2.1, this is also true in the atmosphere, and geostrophic winds may be recognized on weather maps by the fact that the wind-direction arrows are parallel to relatively straight isobars. Even when the motion of the air is strongly curved, and centripetal forces are important (as in the centres of cyclones and anticyclones), wind-direction arrows do not cross the isobars at right angles - which would be directly down the pressure g r a d i e n t - but obliquely (see Figure 2.5 and associated discussion).

So far, we have only been considering forces in a horizontal plane. Figure 3.13 is a partially completed diagram showing the forces acting on a parcel of water of mass m, in a region of the ocean where isobaric surfaces make an angle 0 (greatly exaggerated) with the horizontal. If equilibrium has been attained so that the current is steady and not accelerating, the forces acting in the horizontal direction must balance one another, as must those acting in a vertical direction. The vertical arrow labelled mg represents the weight of the parcel of water, and the two horizontal arrows represent the horizontal pressure gradient force and the Coriolis force.

51

Figure 3.13 Cross-section showing the forces acting on a parcel of water, of mass rn and weight mg, in a region of the ocean where the isobars make an angle 0 (greatly exaggerated) with the horizontal. (Note that if the parcel of water is not to move vertically downwards under the influence of gravity, there must be an equal force acting upwards; this is supplied by pressure.)

Important: Ensure that you understand the answer to this question before moving on.

In Section 3.3.1 we considered a somewhat unrealistic ocean in which conditions were barotropic. We deduced that if the sea-surface (and all other isobars down to the bottom) made an angle of 0 with the horizontal, the horizontal pressure gradient force acting on unit mass of seawater is given by g tan 0 (Equation 3.10a). The horizontal pressure gradient force acting on a water parcel of mass m is therefore given by mg tan 0. We also know that the Coriolis force acting on such a water parcel moving with velocity u is n!fu, where f is the Coriolis parameter (Equation 3.2). In conditions of geostrophic equilibrium, the horizontal pressure gradient force and the Coriolis force balance one another, and we can therefore write:

mg tan 0 = mj5~

or tan O = fit (3.11)

g

Equation 3.11 is known as the g r a d i e n t equation, and in geostrophic flow is true f o r every isobaric surface (Figure 3.14).

It is worth noting that the Coriolis force and the horizontal pressure gradient force are extremely small: they are generally less than 10 -4 N kg -~ and therefore several orders of magnitude smaller than the forces acting in a vertical direction. Nevertheless, in much of the ocean, the Coriolis force and the horizontal pressure gradient force are the largest forces acting in a horizontal direction.

As shown in Figure 3.11 (a), in barotropic conditions the isobaric surfaces follow the sea-surface, even at depth; in baroclinic conditions, by contrast, the extent to which isobaric surfaces follow the sea-surface decreases with increasing depth, as the density distribution has more and more effect. Thus near the surface in Figure 3.11 (b), pressure at a given horizontal level is greater on the left-hand side because the sea-surface is higher and there is a longer column of seawater weighing down above the level in question.

However, water on the right-hand side is more dense, and with increasing depth the greater density increasingly compensates for the lower sea- surface, and the pressures on the two sides become more and more similar.

Figure 3.14 Diagram to illustrate the dynamic equilibrium embodied in the gradient equation.

In geostrophic flow, the horizontal pressure gradient force (mg tan e) is balanced by the equal and opposite Coriolis force (mfu). If you wish to verify for yourself trigonometrically that the horizontal pressure gradient force is given by mg tan e, you will find it helpful to regard the horizontal pressure gradient force as the horizontal component of the resultant pressure which acts at right angles to the isobars; the vertical component is of magnitude mg, balancing the weight of the parcel of water (cf. Figure 3.13).

BAROTROPIC CONDITIONS

(ISOBARIC AND ISOPYCNIC SURFACES PARALLEL)

As geostrophic velocity, u, is proportional to tan t9, the smaller the slope of the isobars, the smaller the geostrophic velocity. If tan e becomes z e r o - i.e.

isobaric surfaces become h o r i z o n t a l - the geostrophic velocity will also be zero. This contrasts with the situation in barotropic conditions, where the geostrophic velocity remains constant with depth.

Figure 3.15 summarizes the differences between barotropic and baroclinic conditions, and illustrates for the two cases how the distribution of density affects the slopes of the isobaric surfaces and how these in turn affect the variation of the geostrophic current velocity with depth.

BAROCLINIC CONDITIONS

(ISOBARIC AND ISOPYCNIC SURFACES INCLINED)

Figure 3.15 Diagrams to summarize the difference between (a) barotropic and (b) baroclinic conditions. The intensity of blue shading corresponds to the density of the water, and the broad arrows indicate the strength of the geostrophic current.

(a) In barotropic flow, isopycnic surfaces (surfaces of constant density) and isobaric surfaces are parallel and their slopes remain constant with depth, because the average density of columns A and B is the same. As the horizontal pressure gradient from B to A is constant with depth, so is the geostrophic current at right angles to it.

(b) In baroclinic flow, the isopycnic surfaces intersect (or are inclined to) isobaric surfaces.

At shallow depths, isobaric surfaces are parallel to the sea-surface, but with increasing depth their slope becomes smaller, because the average density of a column of water at A is more than that of a column of water at B, and with increasing depth this compensates more and more for the effect of the sloping sea-surface. As the isobaric surfaces become increasingly near horizontal, so the horizontal pressure gradient decreases and so does the geostrophic current, until at some depth the isobaric surface is horizontal and the geostrophic current is zero.

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.