## Question

(a I Given the direction of slope of the isobars on Figure 3.13, which of the horizontal arrows represents the horizontal pressure gradient force and which the Coriolis force?

ih> It the current is flowing 'into the page', is the situation illustrated in Figure 3.13 in the Northern or Southern Hemisphere?

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 mfu, where/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:

Equation 3.11 is known as the gradient equation, and in geostrophic flow is true for 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 lO^Nkg-1 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 (mgtan 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).

At some depth, the isobaric surface may well become horizontal. What implication does this have for the velocity of the geostrophic current, given the relationship between isobaric slope and geostrophic velocity, u, expressed b_v ihe gradient equation (Equation 3.11 >?

As geostrophic velocity. //. is proportional to tan 9. the smaller the slope of the isobars, the smaller the geostrophic velocity. If tan 9 becomes zero - i.e. isobaric surfaces become horizontal - 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. isobaric surfaces isopycnic surfaces

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. isobaric surfaces isopycnic surfaces isobaric surfaces isopycnic surfaces