Chapter 10 - Geostrophic Currents

Chapter 10 Contents

10.7 Currents From Hydrographic Sections

Lines of hydrographic data along ship tracks are often used to produce contour plots of density in a vertical section along the track. Cross-sections of currents sometimes show sharply dipping density surfaces with a large contrast in density on either side of the current. The baroclinic currents in the section can be estimated using a technique first proposed by Margules (1906) and described by Defant (1961: Chapter 14). The technique allows oceanographers to estimate the speed and direction of currents perpendicular to the section by a quick look at the section.

Figure 10.11 Slopes b of the sea surface and the slope g of the interface between two homogeneous, moving layers, with density r1 and r2 in the northern hemisphere. From Neumann and Pierson (1966).

To derive Margulesí equation, consider the slope z/x of the stationary interface between two water masses with densities ρ1 and ρ2 (see Figure 10.11). To calculate the change in velocity across the interface we assume homogeneous layers of density ρ1 < ρ2 both of which are in geostrophic equilibrium. Although the ocean does not have an idealized interface that we assumed, and the water masses do not have uniform density, and the interface between the water masses is not sharp, the concept is still useful in practice.

The change in pressure on the interface is:

(10.18)

and the vertical and horizontal pressure gradients are obtained from (10.6):

(10.19)

Therefore:

(10.20a)
(10.20b)

The boundary conditions require δp1 =δp2 on the interface if the boundary is not moving. Equating (10.20a) with (10.20b), dividing by δx, and solving for δz/δx gives:

(10.21a)
(10.21b)
(10.21c)

where β is the slope of the sea surface, and γ is the slope of the boundary between the two water masses. Because the internal differences in density are small, the slope is approximately 1000 times larger than the slope of the constant pressure surfaces. For small values of β and γ, β is the slope of the sea surface, and γ is the slope of the interface.

Consider the application of the technique to the Gulf Stream (Figure 10.8). From the figure: φ = 36°, ρ1 = 1026.7 kg/m3, and ρ2 = 1027.5 kg/m3 at a depth of 500 decibars. If we use the σt = 27.1 surface to estimate the slope between the two water masses, we see that the surface changes from a depth of 350 m to a depth of 650 m over a distance of 70 km. Therefore, tan γ = 4300 ◊ 10-6 = 0.0043, and Δv = v2 - v1 = -0.38 m/s. Assuming v2 = 0, then v1 = 0.38 m/s. This rough estimate of the velocity of the Gulf Stream compares well with velocity at a depth of 500 m calculated from hydrographic data (Table 10.4) assuming a level of no motion at 2,000 decibars.

The slope of the constant-density surfaces are clearly seen in Figure 10.8. And plots of constant-density surfaces can be used to quickly estimate current directions and a rough value for the speed. In contrast, the slope of the sea surface is 8.4 ◊ 10-6 or 0.84 m in 100 km if we use data from Table 10.4.

Note that constant-density surfaces in the Gulf Stream slope downward to the east, and that sea-surface topography slopes upward to the east. Constant pressure and constant density surfaces have opposite slope.

If the sharp interface between two water masses reaches the surface, it is an oceanic front. Such fronts have properties that are very similar to atmospheric fronts.

Eddies in the vicinity of the Gulf Stream can have warm or cold cores (Figure 10.12). Application of Margulesí method these mesoscale eddies gives the direction of the flow. Anticyclonic eddies (clockwise rotation in the northern hemisphere) have warm cores (ρ1 is deeper in the center of the eddy than elsewhere) and the constant-pressure surfaces bow upward. In particular, the sea surface is higher at the center of the ring. Cyclonic eddies are the reverse.

Figure 10.12 Shape of constant-pressure surfaces pi and the interface between two water masses of density r1, r2 if the upper is rotating faster than the lower. Left: Anticyclonic motion, warm-core eddy. Right: Cyclonic, cold-core eddy. Note that the sea surface p0 slopes up toward the center of the warm-core ring, and the constant-density surfaces slope down toward the center. From Defant (1929).

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